id	sid	tid	token	lemma	pos
ejde-942	1	1	electronic	electronic	ADJ
ejde-942	1	2	journal	journal	NOUN
ejde-942	1	3	of	of	ADP
ejde-942	1	4	differential	differential	ADJ
ejde-942	1	5	equations	equation	NOUN
ejde-942	1	6	,	,	PUNCT
ejde-942	1	7	vol	vol	NOUN
ejde-942	1	8	.	.	NOUN
ejde-942	1	9	2024	2024	NUM
ejde-942	1	10	(	(	PUNCT
ejde-942	1	11	2024	2024	NUM
ejde-942	1	12	)	)	PUNCT
ejde-942	1	13	,	,	PUNCT
ejde-942	1	14	no	no	INTJ
ejde-942	1	15	.	.	NOUN
ejde-942	1	16	53	53	NUM
ejde-942	1	17	,	,	PUNCT
ejde-942	1	18	pp	pp	ADJ
ejde-942	1	19	.	.	PUNCT
ejde-942	2	1	1–17	1–17	NOUN
ejde-942	2	2	.	.	PUNCT
ejde-942	3	1	issn	issn	PROPN
ejde-942	3	2	:	:	PUNCT
ejde-942	3	3	1072	1072	NUM
ejde-942	3	4	-	-	SYM
ejde-942	3	5	6691	6691	NUM
ejde-942	3	6	.	.	PUNCT
ejde-942	4	1	url	url	PROPN
ejde-942	4	2	:	:	PUNCT
ejde-942	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-942	4	4	,	,	PUNCT
ejde-942	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-942	4	6	doi	doi	PROPN
ejde-942	4	7	:	:	PUNCT
ejde-942	4	8	10.58997	10.58997	NUM
ejde-942	4	9	/	/	SYM
ejde-942	4	10	ejde.2024.53	ejde.2024.53	NOUN
ejde-942	4	11	deep	deep	ADJ
ejde-942	4	12	learning	learning	NOUN
ejde-942	4	13	method	method	NOUN
ejde-942	4	14	for	for	ADP
ejde-942	4	15	finding	find	VERB
ejde-942	4	16	eigenpairs	eigenpair	NOUN
ejde-942	4	17	in	in	ADP
ejde-942	4	18	sturm	sturm	PROPN
ejde-942	4	19	-	-	PUNCT
ejde-942	4	20	liouville	liouville	NOUN
ejde-942	4	21	eigenvalue	eigenvalue	NOUN
ejde-942	4	22	problems	problem	NOUN
ejde-942	4	23	sen	sen	PROPN
ejde-942	4	24	zhang	zhang	PROPN
ejde-942	4	25	,	,	PUNCT
ejde-942	4	26	jian	jian	PROPN
ejde-942	4	27	zu	zu	PROPN
ejde-942	4	28	,	,	PUNCT
ejde-942	4	29	jingqi	jingqi	NOUN
ejde-942	4	30	zhang	zhang	PROPN
ejde-942	4	31	abstract	abstract	PROPN
ejde-942	4	32	.	.	PUNCT
ejde-942	5	1	solving	solve	VERB
ejde-942	5	2	the	the	DET
ejde-942	5	3	eigenvalue	eigenvalue	PROPN
ejde-942	5	4	problem	problem	NOUN
ejde-942	5	5	for	for	ADP
ejde-942	5	6	differential	differential	ADJ
ejde-942	5	7	equations	equation	NOUN
ejde-942	5	8	in	in	ADP
ejde-942	5	9	inhomogeneous	inhomogeneous	ADJ
ejde-942	5	10	media	medium	NOUN
ejde-942	5	11	poses	pose	VERB
ejde-942	5	12	a	a	DET
ejde-942	5	13	significant	significant	ADJ
ejde-942	5	14	challenge	challenge	NOUN
ejde-942	5	15	across	across	ADP
ejde-942	5	16	diverse	diverse	ADJ
ejde-942	5	17	scientific	scientific	ADJ
ejde-942	5	18	fields	field	NOUN
ejde-942	5	19	.	.	PUNCT
ejde-942	6	1	while	while	SCONJ
ejde-942	6	2	classical	classical	ADJ
ejde-942	6	3	finite	finite	ADJ
ejde-942	6	4	difference	difference	NOUN
ejde-942	6	5	methods	method	NOUN
ejde-942	6	6	and	and	CCONJ
ejde-942	6	7	finite	finite	ADJ
ejde-942	6	8	element	element	NOUN
ejde-942	6	9	methods	method	NOUN
ejde-942	6	10	have	have	AUX
ejde-942	6	11	produced	produce	VERB
ejde-942	6	12	numerous	numerous	ADJ
ejde-942	6	13	outcomes	outcome	NOUN
ejde-942	6	14	,	,	PUNCT
ejde-942	6	15	they	they	PRON
ejde-942	6	16	heavily	heavily	ADV
ejde-942	6	17	rely	rely	VERB
ejde-942	6	18	on	on	ADP
ejde-942	6	19	discretizing	discretize	VERB
ejde-942	6	20	the	the	DET
ejde-942	6	21	computational	computational	ADJ
ejde-942	6	22	domain	domain	NOUN
ejde-942	6	23	,	,	PUNCT
ejde-942	6	24	which	which	PRON
ejde-942	6	25	can	can	AUX
ejde-942	6	26	introduce	introduce	VERB
ejde-942	6	27	complexities	complexity	NOUN
ejde-942	6	28	and	and	CCONJ
ejde-942	6	29	limitations	limitation	NOUN
ejde-942	6	30	.	.	PUNCT
ejde-942	7	1	in	in	ADP
ejde-942	7	2	this	this	DET
ejde-942	7	3	study	study	NOUN
ejde-942	7	4	,	,	PUNCT
ejde-942	7	5	we	we	PRON
ejde-942	7	6	present	present	VERB
ejde-942	7	7	an	an	DET
ejde-942	7	8	unsupervised	unsupervised	ADJ
ejde-942	7	9	neural	neural	ADJ
ejde-942	7	10	network	network	NOUN
ejde-942	7	11	approach	approach	NOUN
ejde-942	7	12	tailored	tailor	VERB
ejde-942	7	13	for	for	ADP
ejde-942	7	14	finding	find	VERB
ejde-942	7	15	eigenpairs	eigenpair	NOUN
ejde-942	7	16	in	in	ADP
ejde-942	7	17	sturm	sturm	PROPN
ejde-942	7	18	-	-	PUNCT
ejde-942	7	19	liouville	liouville	NOUN
ejde-942	7	20	eigenvalue	eigenvalue	NOUN
ejde-942	7	21	problems	problem	NOUN
ejde-942	7	22	within	within	ADP
ejde-942	7	23	inhomogeneous	inhomogeneous	ADJ
ejde-942	7	24	media	medium	NOUN
ejde-942	7	25	.	.	PUNCT
ejde-942	8	1	our	our	PRON
ejde-942	8	2	method	method	NOUN
ejde-942	8	3	introduces	introduce	NOUN
ejde-942	8	4	eigenvalues	eigenvalue	VERB
ejde-942	8	5	as	as	ADP
ejde-942	8	6	trainable	trainable	ADJ
ejde-942	8	7	parameters	parameter	NOUN
ejde-942	8	8	,	,	PUNCT
ejde-942	8	9	crafts	craft	VERB
ejde-942	8	10	a	a	DET
ejde-942	8	11	novel	novel	ADJ
ejde-942	8	12	cost	cost	NOUN
ejde-942	8	13	function	function	NOUN
ejde-942	8	14	,	,	PUNCT
ejde-942	8	15	incorporates	incorporate	VERB
ejde-942	8	16	an	an	DET
ejde-942	8	17	adaptive	adaptive	ADJ
ejde-942	8	18	hyper	hyper	ADJ
ejde-942	8	19	-	-	ADJ
ejde-942	8	20	parameter	parameter	ADJ
ejde-942	8	21	tuning	tuning	NOUN
ejde-942	8	22	strategy	strategy	NOUN
ejde-942	8	23	,	,	PUNCT
ejde-942	8	24	and	and	CCONJ
ejde-942	8	25	sequentially	sequentially	ADV
ejde-942	8	26	trains	train	VERB
ejde-942	8	27	the	the	DET
ejde-942	8	28	eigenpairs	eigenpair	NOUN
ejde-942	8	29	.	.	PUNCT
ejde-942	9	1	the	the	DET
ejde-942	9	2	simplicity	simplicity	NOUN
ejde-942	9	3	,	,	PUNCT
ejde-942	9	4	accuracy	accuracy	NOUN
ejde-942	9	5	,	,	PUNCT
ejde-942	9	6	and	and	CCONJ
ejde-942	9	7	interpretability	interpretability	NOUN
ejde-942	9	8	of	of	ADP
ejde-942	9	9	our	our	PRON
ejde-942	9	10	approach	approach	NOUN
ejde-942	9	11	significantly	significantly	ADV
ejde-942	9	12	expand	expand	VERB
ejde-942	9	13	its	its	PRON
ejde-942	9	14	applicability	applicability	NOUN
ejde-942	9	15	across	across	ADP
ejde-942	9	16	various	various	ADJ
ejde-942	9	17	domains	domain	NOUN
ejde-942	9	18	.	.	PUNCT
ejde-942	10	1	the	the	DET
ejde-942	10	2	method	method	NOUN
ejde-942	10	3	we	we	PRON
ejde-942	10	4	present	present	VERB
ejde-942	10	5	in	in	ADP
ejde-942	10	6	this	this	DET
ejde-942	10	7	paper	paper	NOUN
ejde-942	10	8	can	can	AUX
ejde-942	10	9	easily	easily	ADV
ejde-942	10	10	tackle	tackle	VERB
ejde-942	10	11	boundary	boundary	ADJ
ejde-942	10	12	value	value	NOUN
ejde-942	10	13	conditions	condition	NOUN
ejde-942	10	14	with	with	ADP
ejde-942	10	15	derivatives	derivative	NOUN
ejde-942	10	16	,	,	PUNCT
ejde-942	10	17	resulting	result	VERB
ejde-942	10	18	in	in	ADP
ejde-942	10	19	orthogonal	orthogonal	ADJ
ejde-942	10	20	eigenfunctions	eigenfunction	NOUN
ejde-942	10	21	.	.	PUNCT
ejde-942	11	1	this	this	PRON
ejde-942	11	2	is	be	AUX
ejde-942	11	3	a	a	DET
ejde-942	11	4	very	very	ADV
ejde-942	11	5	important	important	ADJ
ejde-942	11	6	advantage	advantage	NOUN
ejde-942	11	7	of	of	ADP
ejde-942	11	8	deep	deep	ADJ
ejde-942	11	9	learning	learning	NOUN
ejde-942	11	10	methods	method	NOUN
ejde-942	11	11	that	that	PRON
ejde-942	11	12	has	have	AUX
ejde-942	11	13	not	not	PART
ejde-942	11	14	yet	yet	ADV
ejde-942	11	15	been	be	AUX
ejde-942	11	16	noticed	notice	VERB
ejde-942	11	17	.	.	PUNCT
ejde-942	12	1	quantitative	quantitative	ADJ
ejde-942	12	2	estimation	estimation	NOUN
ejde-942	12	3	of	of	ADP
ejde-942	12	4	eigenpairs	eigenpair	NOUN
ejde-942	12	5	is	be	AUX
ejde-942	12	6	given	give	VERB
ejde-942	12	7	for	for	ADP
ejde-942	12	8	the	the	DET
ejde-942	12	9	sturm	sturm	NOUN
ejde-942	12	10	-	-	PUNCT
ejde-942	12	11	liouville	liouville	NOUN
ejde-942	12	12	eigenvalue	eigenvalue	NOUN
ejde-942	12	13	problems	problem	NOUN
ejde-942	12	14	.	.	PUNCT
ejde-942	13	1	furthermore	furthermore	ADV
ejde-942	13	2	,	,	PUNCT
ejde-942	13	3	we	we	PRON
ejde-942	13	4	extend	extend	VERB
ejde-942	13	5	the	the	DET
ejde-942	13	6	proposed	propose	VERB
ejde-942	13	7	methodology	methodology	NOUN
ejde-942	13	8	to	to	PART
ejde-942	13	9	tackle	tackle	VERB
ejde-942	13	10	two	two	NUM
ejde-942	13	11	-	-	PUNCT
ejde-942	13	12	dimensional	dimensional	ADJ
ejde-942	13	13	cases	case	NOUN
ejde-942	13	14	,	,	PUNCT
ejde-942	13	15	periodic	periodic	ADJ
ejde-942	13	16	scenarios	scenario	NOUN
ejde-942	13	17	,	,	PUNCT
ejde-942	13	18	demonstrating	demonstrate	VERB
ejde-942	13	19	its	its	PRON
ejde-942	13	20	versatility	versatility	NOUN
ejde-942	13	21	and	and	CCONJ
ejde-942	13	22	broad	broad	ADJ
ejde-942	13	23	potential	potential	NOUN
ejde-942	13	24	.	.	PUNCT
ejde-942	14	1	1	1	X
ejde-942	14	2	.	.	X
ejde-942	14	3	introduction	introduction	NOUN
ejde-942	14	4	the	the	DET
ejde-942	14	5	sturm	sturm	PROPN
ejde-942	14	6	-	-	PUNCT
ejde-942	14	7	liouville	liouville	VERB
ejde-942	14	8	eigenvalue	eigenvalue	NOUN
ejde-942	14	9	problem	problem	NOUN
ejde-942	14	10	holds	hold	VERB
ejde-942	14	11	substantial	substantial	ADJ
ejde-942	14	12	significance	significance	NOUN
ejde-942	14	13	across	across	ADP
ejde-942	14	14	a	a	DET
ejde-942	14	15	diverse	diverse	ADJ
ejde-942	14	16	range	range	NOUN
ejde-942	14	17	of	of	ADP
ejde-942	14	18	fields	field	NOUN
ejde-942	14	19	.	.	PUNCT
ejde-942	15	1	it	it	PRON
ejde-942	15	2	can	can	AUX
ejde-942	15	3	be	be	AUX
ejde-942	15	4	demonstrated	demonstrate	VERB
ejde-942	15	5	that	that	SCONJ
ejde-942	15	6	there	there	PRON
ejde-942	15	7	are	be	VERB
ejde-942	15	8	an	an	DET
ejde-942	15	9	infinite	infinite	ADJ
ejde-942	15	10	number	number	NOUN
ejde-942	15	11	of	of	ADP
ejde-942	15	12	eigenvalues	eigenvalue	NOUN
ejde-942	15	13	,	,	PUNCT
ejde-942	15	14	each	each	PRON
ejde-942	15	15	with	with	ADP
ejde-942	15	16	a	a	DET
ejde-942	15	17	unique	unique	ADJ
ejde-942	15	18	eigenfunction	eigenfunction	NOUN
ejde-942	15	19	,	,	PUNCT
ejde-942	15	20	and	and	CCONJ
ejde-942	15	21	that	that	SCONJ
ejde-942	15	22	these	these	DET
ejde-942	15	23	eigenfunctions	eigenfunction	NOUN
ejde-942	15	24	form	form	VERB
ejde-942	15	25	an	an	DET
ejde-942	15	26	orthonormal	orthonormal	ADJ
ejde-942	15	27	basis	basis	NOUN
ejde-942	15	28	within	within	ADP
ejde-942	15	29	a	a	DET
ejde-942	15	30	specific	specific	ADJ
ejde-942	15	31	hilbert	hilbert	NOUN
ejde-942	15	32	space	space	NOUN
ejde-942	15	33	of	of	ADP
ejde-942	15	34	functions	function	NOUN
ejde-942	15	35	.	.	PUNCT
ejde-942	16	1	the	the	DET
ejde-942	16	2	adaptability	adaptability	NOUN
ejde-942	16	3	of	of	ADP
ejde-942	16	4	the	the	DET
ejde-942	16	5	sturm	sturm	NOUN
ejde-942	16	6	-	-	PUNCT
ejde-942	16	7	liouville	liouville	NOUN
ejde-942	16	8	equation	equation	NOUN
ejde-942	16	9	to	to	ADP
ejde-942	16	10	various	various	ADJ
ejde-942	16	11	boundary	boundary	ADJ
ejde-942	16	12	value	value	NOUN
ejde-942	16	13	conditions	condition	NOUN
ejde-942	16	14	allows	allow	VERB
ejde-942	16	15	for	for	ADP
ejde-942	16	16	the	the	DET
ejde-942	16	17	precise	precise	ADJ
ejde-942	16	18	representation	representation	NOUN
ejde-942	16	19	of	of	ADP
ejde-942	16	20	specific	specific	ADJ
ejde-942	16	21	physical	physical	ADJ
ejde-942	16	22	phenomena	phenomenon	NOUN
ejde-942	16	23	.	.	PUNCT
ejde-942	17	1	for	for	ADP
ejde-942	17	2	example	example	NOUN
ejde-942	17	3	,	,	PUNCT
ejde-942	17	4	in	in	ADP
ejde-942	17	5	the	the	DET
ejde-942	17	6	context	context	NOUN
ejde-942	17	7	of	of	ADP
ejde-942	17	8	heat	heat	NOUN
ejde-942	17	9	equations	equation	NOUN
ejde-942	17	10	,	,	PUNCT
ejde-942	17	11	dirichlet	dirichlet	PROPN
ejde-942	17	12	conditions	condition	NOUN
ejde-942	17	13	maintain	maintain	VERB
ejde-942	17	14	a	a	DET
ejde-942	17	15	consistent	consistent	ADJ
ejde-942	17	16	temperature	temperature	NOUN
ejde-942	17	17	at	at	ADP
ejde-942	17	18	the	the	DET
ejde-942	17	19	rod	rod	NOUN
ejde-942	17	20	’s	’s	PART
ejde-942	17	21	extremities	extremity	NOUN
ejde-942	17	22	,	,	PUNCT
ejde-942	17	23	while	while	SCONJ
ejde-942	17	24	neumann	neumann	PROPN
ejde-942	17	25	conditions	condition	NOUN
ejde-942	17	26	simulate	simulate	VERB
ejde-942	17	27	insulating	insulate	VERB
ejde-942	17	28	environments	environment	NOUN
ejde-942	17	29	with	with	ADP
ejde-942	17	30	no	no	DET
ejde-942	17	31	heat	heat	NOUN
ejde-942	17	32	transfer	transfer	NOUN
ejde-942	17	33	and	and	CCONJ
ejde-942	17	34	negligible	negligible	ADJ
ejde-942	17	35	temperature	temperature	NOUN
ejde-942	17	36	gradients	gradient	NOUN
ejde-942	17	37	at	at	ADP
ejde-942	17	38	the	the	DET
ejde-942	17	39	ends	end	NOUN
ejde-942	17	40	.	.	PUNCT
ejde-942	18	1	furthermore	furthermore	ADV
ejde-942	18	2	,	,	PUNCT
ejde-942	18	3	more	more	ADV
ejde-942	18	4	comprehensive	comprehensive	ADJ
ejde-942	18	5	boundary	boundary	ADJ
ejde-942	18	6	value	value	NOUN
ejde-942	18	7	conditions	condition	NOUN
ejde-942	18	8	,	,	PUNCT
ejde-942	18	9	such	such	ADJ
ejde-942	18	10	as	as	ADP
ejde-942	18	11	the	the	DET
ejde-942	18	12	robin	robin	PROPN
ejde-942	18	13	condition	condition	NOUN
ejde-942	18	14	,	,	PUNCT
ejde-942	18	15	accommodate	accommodate	NOUN
ejde-942	18	16	partially	partially	ADV
ejde-942	18	17	insulated	insulate	VERB
ejde-942	18	18	boundaries	boundary	NOUN
ejde-942	18	19	.	.	PUNCT
ejde-942	19	1	in	in	ADP
ejde-942	19	2	practical	practical	ADJ
ejde-942	19	3	applications	application	NOUN
ejde-942	19	4	,	,	PUNCT
ejde-942	19	5	it	it	PRON
ejde-942	19	6	is	be	AUX
ejde-942	19	7	also	also	ADV
ejde-942	19	8	crucial	crucial	ADJ
ejde-942	19	9	to	to	PART
ejde-942	19	10	compute	compute	VERB
ejde-942	19	11	multiple	multiple	ADJ
ejde-942	19	12	orthogonal	orthogonal	ADJ
ejde-942	19	13	eigenvectors	eigenvector	NOUN
ejde-942	19	14	of	of	ADP
ejde-942	19	15	the	the	DET
ejde-942	19	16	sturm	sturm	NOUN
ejde-942	19	17	-	-	PUNCT
ejde-942	19	18	liouville	liouville	NOUN
ejde-942	19	19	eigenvalue	eigenvalue	NOUN
ejde-942	19	20	problem	problem	NOUN
ejde-942	19	21	.	.	PUNCT
ejde-942	20	1	for	for	ADP
ejde-942	20	2	instance	instance	NOUN
ejde-942	20	3	,	,	PUNCT
ejde-942	20	4	in	in	ADP
ejde-942	20	5	the	the	DET
ejde-942	20	6	construction	construction	NOUN
ejde-942	20	7	2020	2020	NUM
ejde-942	20	8	mathematics	mathematic	NOUN
ejde-942	20	9	subject	subject	ADJ
ejde-942	20	10	classification	classification	NOUN
ejde-942	20	11	.	.	PUNCT
ejde-942	20	12	34l16	34l16	NUM
ejde-942	20	13	,	,	PUNCT
ejde-942	20	14	65f18	65f18	NUM
ejde-942	20	15	,	,	PUNCT
ejde-942	20	16	65l15	65l15	NUM
ejde-942	20	17	,	,	PUNCT
ejde-942	20	18	65n25	65n25	NOUN
ejde-942	20	19	,	,	PUNCT
ejde-942	20	20	68t07	68t07	NOUN
ejde-942	20	21	.	.	PUNCT
ejde-942	21	1	key	key	ADJ
ejde-942	21	2	words	word	NOUN
ejde-942	21	3	and	and	CCONJ
ejde-942	21	4	phrases	phrase	NOUN
ejde-942	21	5	.	.	PUNCT
ejde-942	22	1	eigenvalue	eigenvalue	NOUN
ejde-942	22	2	problems	problem	NOUN
ejde-942	22	3	;	;	PUNCT
ejde-942	22	4	sturm	sturm	NOUN
ejde-942	22	5	-	-	PUNCT
ejde-942	22	6	liouville	liouville	NOUN
ejde-942	22	7	problems	problem	NOUN
ejde-942	22	8	;	;	PUNCT
ejde-942	22	9	neural	neural	ADJ
ejde-942	22	10	network	network	NOUN
ejde-942	22	11	;	;	PUNCT
ejde-942	22	12	self	self	NOUN
ejde-942	22	13	-	-	PUNCT
ejde-942	22	14	adjoint	adjoint	NOUN
ejde-942	22	15	operator	operator	NOUN
ejde-942	22	16	;	;	PUNCT
ejde-942	22	17	inhomogeneous	inhomogeneous	ADJ
ejde-942	22	18	media	medium	NOUN
ejde-942	22	19	.	.	PUNCT
ejde-942	23	1	©	©	ADP
ejde-942	23	2	2024	2024	NUM
ejde-942	23	3	.	.	PUNCT
ejde-942	24	1	this	this	DET
ejde-942	24	2	work	work	NOUN
ejde-942	24	3	is	be	AUX
ejde-942	24	4	licensed	license	VERB
ejde-942	24	5	under	under	ADP
ejde-942	24	6	a	a	DET
ejde-942	24	7	cc	cc	NOUN
ejde-942	24	8	by	by	ADP
ejde-942	24	9	4.0	4.0	NUM
ejde-942	24	10	license	license	NOUN
ejde-942	24	11	.	.	PUNCT
ejde-942	25	1	submitted	submit	VERB
ejde-942	25	2	december	december	PROPN
ejde-942	25	3	2	2	NUM
ejde-942	25	4	,	,	PUNCT
ejde-942	25	5	2023	2023	NUM
ejde-942	25	6	.	.	PUNCT
ejde-942	26	1	published	publish	VERB
ejde-942	26	2	september	september	PROPN
ejde-942	26	3	12	12	NUM
ejde-942	26	4	,	,	PUNCT
ejde-942	26	5	2024	2024	NUM
ejde-942	26	6	.	.	PUNCT
ejde-942	26	7	1	1	NUM
ejde-942	26	8	2	2	NUM
ejde-942	26	9	s.	s.	PROPN
ejde-942	26	10	zhang	zhang	PROPN
ejde-942	26	11	,	,	PUNCT
ejde-942	26	12	j.	j.	PROPN
ejde-942	26	13	zu	zu	PROPN
ejde-942	26	14	,	,	PUNCT
ejde-942	26	15	j.	j.	PROPN
ejde-942	26	16	zhang	zhang	PROPN
ejde-942	26	17	ejde-2024/53	ejde-2024/53	PROPN
ejde-942	26	18	of	of	ADP
ejde-942	26	19	a	a	DET
ejde-942	26	20	suitable	suitable	ADJ
ejde-942	26	21	lyapunov	lyapunov	NOUN
ejde-942	26	22	function	function	NOUN
ejde-942	26	23	,	,	PUNCT
ejde-942	26	24	there	there	PRON
ejde-942	26	25	are	be	VERB
ejde-942	26	26	instances	instance	NOUN
ejde-942	26	27	where	where	SCONJ
ejde-942	26	28	the	the	DET
ejde-942	26	29	need	need	NOUN
ejde-942	26	30	for	for	ADP
ejde-942	26	31	several	several	ADJ
ejde-942	26	32	mutually	mutually	ADV
ejde-942	26	33	orthogonal	orthogonal	ADJ
ejde-942	26	34	minimal	minimal	ADJ
ejde-942	26	35	eigenvectors	eigenvector	NOUN
ejde-942	26	36	arises	arise	VERB
ejde-942	26	37	,	,	PUNCT
ejde-942	26	38	see	see	VERB
ejde-942	26	39	[	[	X
ejde-942	26	40	4	4	NUM
ejde-942	26	41	]	]	PUNCT
ejde-942	26	42	.	.	PUNCT
ejde-942	27	1	therefore	therefore	ADV
ejde-942	27	2	,	,	PUNCT
ejde-942	27	3	the	the	DET
ejde-942	27	4	quest	quest	NOUN
ejde-942	27	5	for	for	ADP
ejde-942	27	6	multiple	multiple	ADJ
ejde-942	27	7	minimal	minimal	ADJ
ejde-942	27	8	eigenpairs	eigenpair	NOUN
ejde-942	27	9	(	(	PUNCT
ejde-942	27	10	i.e.	i.e.	X
ejde-942	27	11	,	,	PUNCT
ejde-942	27	12	eigenvalues	eigenvalue	NOUN
ejde-942	27	13	and	and	CCONJ
ejde-942	27	14	eigenvectors	eigenvector	NOUN
ejde-942	27	15	)	)	PUNCT
ejde-942	27	16	for	for	ADP
ejde-942	27	17	the	the	DET
ejde-942	27	18	sturm	sturm	PROPN
ejde-942	27	19	-	-	PUNCT
ejde-942	27	20	liouville	liouville	VERB
ejde-942	27	21	eigenvalue	eigenvalue	NOUN
ejde-942	27	22	problem	problem	NOUN
ejde-942	27	23	remains	remain	VERB
ejde-942	27	24	an	an	DET
ejde-942	27	25	enduring	endure	VERB
ejde-942	27	26	research	research	NOUN
ejde-942	27	27	pursuit	pursuit	NOUN
ejde-942	27	28	.	.	PUNCT
ejde-942	28	1	classical	classical	ADJ
ejde-942	28	2	numerical	numerical	ADJ
ejde-942	28	3	methods	method	NOUN
ejde-942	28	4	employed	employ	VERB
ejde-942	28	5	in	in	ADP
ejde-942	28	6	the	the	DET
ejde-942	28	7	quest	quest	NOUN
ejde-942	28	8	for	for	ADP
ejde-942	28	9	eigenpairs	eigenpair	NOUN
ejde-942	28	10	include	include	VERB
ejde-942	28	11	finite	finite	ADJ
ejde-942	28	12	difference	difference	NOUN
ejde-942	28	13	methods	method	NOUN
ejde-942	28	14	[	[	X
ejde-942	28	15	1	1	NUM
ejde-942	28	16	,	,	PUNCT
ejde-942	28	17	17	17	NUM
ejde-942	28	18	]	]	PUNCT
ejde-942	28	19	and	and	CCONJ
ejde-942	28	20	finite	finite	ADJ
ejde-942	28	21	element	element	NOUN
ejde-942	28	22	methods	method	NOUN
ejde-942	28	23	[	[	X
ejde-942	28	24	2	2	NUM
ejde-942	28	25	]	]	PUNCT
ejde-942	28	26	.	.	PUNCT
ejde-942	29	1	the	the	DET
ejde-942	29	2	finite	finite	ADJ
ejde-942	29	3	difference	difference	NOUN
ejde-942	29	4	method	method	NOUN
ejde-942	29	5	,	,	PUNCT
ejde-942	29	6	characterized	characterize	VERB
ejde-942	29	7	by	by	ADP
ejde-942	29	8	its	its	PRON
ejde-942	29	9	simplicity	simplicity	NOUN
ejde-942	29	10	and	and	CCONJ
ejde-942	29	11	ease	ease	NOUN
ejde-942	29	12	of	of	ADP
ejde-942	29	13	comprehension	comprehension	NOUN
ejde-942	29	14	,	,	PUNCT
ejde-942	29	15	is	be	AUX
ejde-942	29	16	a	a	DET
ejde-942	29	17	popular	popular	ADJ
ejde-942	29	18	choice	choice	NOUN
ejde-942	29	19	.	.	PUNCT
ejde-942	30	1	however	however	ADV
ejde-942	30	2	,	,	PUNCT
ejde-942	30	3	it	it	PRON
ejde-942	30	4	also	also	ADV
ejde-942	30	5	exhibits	exhibit	VERB
ejde-942	30	6	inherent	inherent	ADJ
ejde-942	30	7	limitations	limitation	NOUN
ejde-942	30	8	.	.	PUNCT
ejde-942	31	1	for	for	ADP
ejde-942	31	2	instance	instance	NOUN
ejde-942	31	3	,	,	PUNCT
ejde-942	31	4	in	in	ADP
ejde-942	31	5	the	the	DET
ejde-942	31	6	case	case	NOUN
ejde-942	31	7	of	of	ADP
ejde-942	31	8	sturm	sturm	PROPN
ejde-942	31	9	-	-	PUNCT
ejde-942	31	10	liouville	liouville	NOUN
ejde-942	31	11	eigenvalue	eigenvalue	NOUN
ejde-942	31	12	problems	problem	NOUN
ejde-942	31	13	with	with	ADP
ejde-942	31	14	derivative	derivative	ADJ
ejde-942	31	15	boundary	boundary	ADJ
ejde-942	31	16	value	value	NOUN
ejde-942	31	17	conditions	condition	NOUN
ejde-942	31	18	,	,	PUNCT
ejde-942	31	19	such	such	ADJ
ejde-942	31	20	as	as	ADP
ejde-942	31	21	neumann	neumann	PROPN
ejde-942	31	22	boundary	boundary	PROPN
ejde-942	31	23	and	and	CCONJ
ejde-942	31	24	robin	robin	PROPN
ejde-942	31	25	boundary	boundary	PROPN
ejde-942	31	26	,	,	PUNCT
ejde-942	31	27	the	the	DET
ejde-942	31	28	finite	finite	ADJ
ejde-942	31	29	difference	difference	NOUN
ejde-942	31	30	approximation	approximation	NOUN
ejde-942	31	31	of	of	ADP
ejde-942	31	32	the	the	DET
ejde-942	31	33	differential	differential	ADJ
ejde-942	31	34	operator	operator	NOUN
ejde-942	31	35	may	may	AUX
ejde-942	31	36	not	not	PART
ejde-942	31	37	preserve	preserve	VERB
ejde-942	31	38	self	self	NOUN
ejde-942	31	39	-	-	PUNCT
ejde-942	31	40	adjointness	adjointness	NOUN
ejde-942	31	41	.	.	PUNCT
ejde-942	32	1	consequently	consequently	ADV
ejde-942	32	2	,	,	PUNCT
ejde-942	32	3	the	the	DET
ejde-942	32	4	resulting	result	VERB
ejde-942	32	5	eigenvectors	eigenvector	NOUN
ejde-942	32	6	are	be	AUX
ejde-942	32	7	not	not	PART
ejde-942	32	8	orthogonal	orthogonal	ADJ
ejde-942	32	9	to	to	ADP
ejde-942	32	10	each	each	DET
ejde-942	32	11	other	other	ADJ
ejde-942	32	12	,	,	PUNCT
ejde-942	32	13	which	which	PRON
ejde-942	32	14	contradicts	contradict	VERB
ejde-942	32	15	the	the	DET
ejde-942	32	16	nature	nature	NOUN
ejde-942	32	17	of	of	ADP
ejde-942	32	18	sturm	sturm	NOUN
ejde-942	32	19	-	-	PUNCT
ejde-942	32	20	liouville	liouville	VERB
ejde-942	32	21	differential	differential	NOUN
ejde-942	32	22	operators	operator	NOUN
ejde-942	32	23	being	be	AUX
ejde-942	32	24	self	self	NOUN
ejde-942	32	25	-	-	PUNCT
ejde-942	32	26	adjoint	adjoint	NOUN
ejde-942	32	27	and	and	CCONJ
ejde-942	32	28	having	have	VERB
ejde-942	32	29	orthogonal	orthogonal	ADJ
ejde-942	32	30	eigenvectors	eigenvector	NOUN
ejde-942	32	31	.	.	PUNCT
ejde-942	33	1	the	the	DET
ejde-942	33	2	finite	finite	PROPN
ejde-942	33	3	element	element	NOUN
ejde-942	33	4	method	method	NOUN
ejde-942	33	5	offers	offer	VERB
ejde-942	33	6	high	high	ADJ
ejde-942	33	7	computational	computational	ADJ
ejde-942	33	8	accuracy	accuracy	NOUN
ejde-942	33	9	and	and	CCONJ
ejde-942	33	10	wide	wide	ADJ
ejde-942	33	11	applicability	applicability	NOUN
ejde-942	33	12	.	.	PUNCT
ejde-942	34	1	however	however	ADV
ejde-942	34	2	,	,	PUNCT
ejde-942	34	3	it	it	PRON
ejde-942	34	4	requires	require	VERB
ejde-942	34	5	manual	manual	ADJ
ejde-942	34	6	specification	specification	NOUN
ejde-942	34	7	of	of	ADP
ejde-942	34	8	appropriate	appropriate	ADJ
ejde-942	34	9	finite	finite	ADJ
ejde-942	34	10	element	element	NOUN
ejde-942	34	11	units	unit	NOUN
ejde-942	34	12	prior	prior	ADV
ejde-942	34	13	to	to	ADP
ejde-942	34	14	computation	computation	NOUN
ejde-942	34	15	,	,	PUNCT
ejde-942	34	16	and	and	CCONJ
ejde-942	34	17	its	its	PRON
ejde-942	34	18	effectiveness	effectiveness	NOUN
ejde-942	34	19	is	be	AUX
ejde-942	34	20	often	often	ADV
ejde-942	34	21	reliant	reliant	ADJ
ejde-942	34	22	on	on	ADP
ejde-942	34	23	individual	individual	ADJ
ejde-942	34	24	expertise	expertise	NOUN
ejde-942	34	25	.	.	PUNCT
ejde-942	35	1	it	it	PRON
ejde-942	35	2	is	be	AUX
ejde-942	35	3	a	a	DET
ejde-942	35	4	more	more	ADV
ejde-942	35	5	complex	complex	ADJ
ejde-942	35	6	method	method	NOUN
ejde-942	35	7	and	and	CCONJ
ejde-942	35	8	may	may	AUX
ejde-942	35	9	not	not	PART
ejde-942	35	10	be	be	AUX
ejde-942	35	11	user	user	NOUN
ejde-942	35	12	-	-	PUNCT
ejde-942	35	13	friendly	friendly	ADJ
ejde-942	35	14	for	for	ADP
ejde-942	35	15	non	non	NOUN
ejde-942	35	16	-	-	NOUN
ejde-942	35	17	experts	expert	NOUN
ejde-942	35	18	.	.	PUNCT
ejde-942	36	1	furthermore	furthermore	ADV
ejde-942	36	2	,	,	PUNCT
ejde-942	36	3	it	it	PRON
ejde-942	36	4	is	be	AUX
ejde-942	36	5	worth	worth	ADJ
ejde-942	36	6	noting	note	VERB
ejde-942	36	7	that	that	SCONJ
ejde-942	36	8	both	both	DET
ejde-942	36	9	finite	finite	ADJ
ejde-942	36	10	difference	difference	NOUN
ejde-942	36	11	methods	method	NOUN
ejde-942	36	12	and	and	CCONJ
ejde-942	36	13	finite	finite	ADJ
ejde-942	36	14	element	element	NOUN
ejde-942	36	15	methods	method	NOUN
ejde-942	36	16	yield	yield	VERB
ejde-942	36	17	numerical	numerical	ADJ
ejde-942	36	18	solutions	solution	NOUN
ejde-942	36	19	,	,	PUNCT
ejde-942	36	20	which	which	PRON
ejde-942	36	21	lack	lack	VERB
ejde-942	36	22	analytical	analytical	ADJ
ejde-942	36	23	expressions	expression	NOUN
ejde-942	36	24	and	and	CCONJ
ejde-942	36	25	can	can	AUX
ejde-942	36	26	not	not	PART
ejde-942	36	27	be	be	AUX
ejde-942	36	28	readily	readily	ADV
ejde-942	36	29	used	use	VERB
ejde-942	36	30	for	for	ADP
ejde-942	36	31	operations	operation	NOUN
ejde-942	36	32	such	such	ADJ
ejde-942	36	33	as	as	ADP
ejde-942	36	34	differentiation	differentiation	NOUN
ejde-942	36	35	.	.	PUNCT
ejde-942	37	1	their	their	PRON
ejde-942	37	2	applicability	applicability	NOUN
ejde-942	37	3	is	be	AUX
ejde-942	37	4	relatively	relatively	ADV
ejde-942	37	5	limited	limited	ADJ
ejde-942	37	6	in	in	ADP
ejde-942	37	7	situations	situation	NOUN
ejde-942	37	8	requiring	require	VERB
ejde-942	37	9	analytical	analytical	ADJ
ejde-942	37	10	expressions	expression	NOUN
ejde-942	37	11	.	.	PUNCT
ejde-942	38	1	the	the	DET
ejde-942	38	2	search	search	NOUN
ejde-942	38	3	for	for	ADP
ejde-942	38	4	effective	effective	ADJ
ejde-942	38	5	analytical	analytical	ADJ
ejde-942	38	6	approximations	approximation	NOUN
ejde-942	38	7	remains	remain	VERB
ejde-942	38	8	a	a	DET
ejde-942	38	9	significant	significant	ADJ
ejde-942	38	10	scientific	scientific	ADJ
ejde-942	38	11	challenge	challenge	NOUN
ejde-942	38	12	.	.	PUNCT
ejde-942	39	1	in	in	ADP
ejde-942	39	2	recent	recent	ADJ
ejde-942	39	3	years	year	NOUN
ejde-942	39	4	,	,	PUNCT
ejde-942	39	5	deep	deep	ADJ
ejde-942	39	6	learning	learning	NOUN
ejde-942	39	7	methods	method	NOUN
ejde-942	39	8	have	have	AUX
ejde-942	39	9	rapidly	rapidly	ADV
ejde-942	39	10	advanced	advance	VERB
ejde-942	39	11	and	and	CCONJ
ejde-942	39	12	integrated	integrate	VERB
ejde-942	39	13	into	into	ADP
ejde-942	39	14	diverse	diverse	ADJ
ejde-942	39	15	aspects	aspect	NOUN
ejde-942	39	16	of	of	ADP
ejde-942	39	17	people	people	NOUN
ejde-942	39	18	’s	’s	PART
ejde-942	39	19	lives	life	NOUN
ejde-942	39	20	.	.	PUNCT
ejde-942	40	1	within	within	ADP
ejde-942	40	2	the	the	DET
ejde-942	40	3	study	study	NOUN
ejde-942	40	4	of	of	ADP
ejde-942	40	5	differential	differential	ADJ
ejde-942	40	6	equations	equation	NOUN
ejde-942	40	7	,	,	PUNCT
ejde-942	40	8	their	their	PRON
ejde-942	40	9	exceptional	exceptional	ADJ
ejde-942	40	10	strengths	strength	NOUN
ejde-942	40	11	and	and	CCONJ
ejde-942	40	12	capabilities	capability	NOUN
ejde-942	40	13	stand	stand	VERB
ejde-942	40	14	out	out	ADP
ejde-942	40	15	.	.	PUNCT
ejde-942	41	1	chen	chen	PROPN
ejde-942	41	2	et	et	PROPN
ejde-942	41	3	al	al	PROPN
ejde-942	41	4	.	.	PUNCT
ejde-942	42	1	[	[	X
ejde-942	42	2	6	6	NUM
ejde-942	42	3	]	]	PUNCT
ejde-942	42	4	proposed	propose	VERB
ejde-942	42	5	neural	neural	ADJ
ejde-942	42	6	ode	ode	PROPN
ejde-942	42	7	,	,	PUNCT
ejde-942	42	8	which	which	PRON
ejde-942	42	9	approximate	approximate	VERB
ejde-942	42	10	the	the	DET
ejde-942	42	11	right	right	ADJ
ejde-942	42	12	-	-	PUNCT
ejde-942	42	13	hand	hand	NOUN
ejde-942	42	14	side	side	NOUN
ejde-942	42	15	function	function	NOUN
ejde-942	42	16	of	of	ADP
ejde-942	42	17	differential	differential	ADJ
ejde-942	42	18	equations	equation	NOUN
ejde-942	42	19	using	use	VERB
ejde-942	42	20	deep	deep	ADJ
ejde-942	42	21	neural	neural	ADJ
ejde-942	42	22	networks	network	NOUN
ejde-942	42	23	,	,	PUNCT
ejde-942	42	24	achieving	achieve	VERB
ejde-942	42	25	remarkable	remarkable	ADJ
ejde-942	42	26	results	result	NOUN
ejde-942	42	27	.	.	PUNCT
ejde-942	43	1	currently	currently	ADV
ejde-942	43	2	,	,	PUNCT
ejde-942	43	3	the	the	DET
ejde-942	43	4	most	most	ADV
ejde-942	43	5	widely	widely	ADV
ejde-942	43	6	used	use	VERB
ejde-942	43	7	approach	approach	NOUN
ejde-942	43	8	is	be	AUX
ejde-942	43	9	to	to	PART
ejde-942	43	10	approximate	approximate	VERB
ejde-942	43	11	the	the	DET
ejde-942	43	12	solutions	solution	NOUN
ejde-942	43	13	of	of	ADP
ejde-942	43	14	differential	differential	ADJ
ejde-942	43	15	equations	equation	NOUN
ejde-942	43	16	using	use	VERB
ejde-942	43	17	deep	deep	ADJ
ejde-942	43	18	neural	neural	ADJ
ejde-942	43	19	networks	network	NOUN
ejde-942	43	20	,	,	PUNCT
ejde-942	43	21	with	with	ADP
ejde-942	43	22	advantages	advantage	NOUN
ejde-942	43	23	like	like	ADP
ejde-942	43	24	increased	increase	VERB
ejde-942	43	25	applicability	applicability	NOUN
ejde-942	43	26	,	,	PUNCT
ejde-942	43	27	ease	ease	NOUN
ejde-942	43	28	of	of	ADP
ejde-942	43	29	understanding	understanding	NOUN
ejde-942	43	30	,	,	PUNCT
ejde-942	43	31	and	and	CCONJ
ejde-942	43	32	the	the	DET
ejde-942	43	33	ability	ability	NOUN
ejde-942	43	34	to	to	PART
ejde-942	43	35	perform	perform	VERB
ejde-942	43	36	differentiable	differentiable	ADJ
ejde-942	43	37	operations	operation	NOUN
ejde-942	43	38	.	.	PUNCT
ejde-942	44	1	raissi	raissi	ADJ
ejde-942	44	2	et	et	PROPN
ejde-942	44	3	al	al	PROPN
ejde-942	44	4	.	.	PUNCT
ejde-942	45	1	[	[	X
ejde-942	45	2	19	19	NUM
ejde-942	45	3	]	]	PUNCT
ejde-942	45	4	introduced	introduce	VERB
ejde-942	45	5	a	a	DET
ejde-942	45	6	deep	deep	ADJ
ejde-942	45	7	learning	learning	NOUN
ejde-942	45	8	framework	framework	NOUN
ejde-942	45	9	known	know	VERB
ejde-942	45	10	as	as	ADP
ejde-942	45	11	’	'	PUNCT
ejde-942	45	12	physics	physics	NOUN
ejde-942	45	13	-	-	PUNCT
ejde-942	45	14	informed	inform	VERB
ejde-942	45	15	neural	neural	ADJ
ejde-942	45	16	networks	network	NOUN
ejde-942	45	17	’	'	PUNCT
ejde-942	45	18	(	(	PUNCT
ejde-942	45	19	pinn	pinn	PROPN
ejde-942	45	20	)	)	PUNCT
ejde-942	45	21	.	.	PUNCT
ejde-942	46	1	within	within	ADP
ejde-942	46	2	this	this	DET
ejde-942	46	3	innovative	innovative	ADJ
ejde-942	46	4	framework	framework	NOUN
ejde-942	46	5	,	,	PUNCT
ejde-942	46	6	the	the	DET
ejde-942	46	7	physical	physical	ADJ
ejde-942	46	8	equations	equation	NOUN
ejde-942	46	9	themselves	themselves	PRON
ejde-942	46	10	are	be	AUX
ejde-942	46	11	seamlessly	seamlessly	ADV
ejde-942	46	12	integrated	integrate	VERB
ejde-942	46	13	into	into	ADP
ejde-942	46	14	the	the	DET
ejde-942	46	15	network	network	NOUN
ejde-942	46	16	as	as	ADP
ejde-942	46	17	constraints	constraint	NOUN
ejde-942	46	18	,	,	PUNCT
ejde-942	46	19	thereby	thereby	ADV
ejde-942	46	20	enabling	enable	VERB
ejde-942	46	21	the	the	DET
ejde-942	46	22	solution	solution	NOUN
ejde-942	46	23	of	of	ADP
ejde-942	46	24	partial	partial	ADJ
ejde-942	46	25	differential	differential	ADJ
ejde-942	46	26	equations	equation	NOUN
ejde-942	46	27	(	(	PUNCT
ejde-942	46	28	pdes	pde	NOUN
ejde-942	46	29	)	)	PUNCT
ejde-942	46	30	without	without	ADP
ejde-942	46	31	the	the	DET
ejde-942	46	32	necessity	necessity	NOUN
ejde-942	46	33	of	of	ADP
ejde-942	46	34	a	a	DET
ejde-942	46	35	dataset	dataset	NOUN
ejde-942	46	36	.	.	PUNCT
ejde-942	47	1	pang	pang	NOUN
ejde-942	47	2	et	et	PROPN
ejde-942	47	3	al	al	PROPN
ejde-942	47	4	.	.	PUNCT
ejde-942	48	1	[	[	X
ejde-942	48	2	18	18	NUM
ejde-942	48	3	]	]	PUNCT
ejde-942	48	4	proposed	propose	VERB
ejde-942	48	5	a	a	DET
ejde-942	48	6	fractional	fractional	ADJ
ejde-942	48	7	pinn	pinn	NOUN
ejde-942	48	8	(	(	PUNCT
ejde-942	48	9	fpinn	fpinn	PROPN
ejde-942	48	10	)	)	PUNCT
ejde-942	48	11	,	,	PUNCT
ejde-942	48	12	addressing	address	VERB
ejde-942	48	13	the	the	DET
ejde-942	48	14	challenge	challenge	NOUN
ejde-942	48	15	of	of	ADP
ejde-942	48	16	automatic	automatic	ADJ
ejde-942	48	17	differentiation	differentiation	NOUN
ejde-942	48	18	being	be	AUX
ejde-942	48	19	unsuitable	unsuitable	ADJ
ejde-942	48	20	for	for	ADP
ejde-942	48	21	fractional	fractional	ADJ
ejde-942	48	22	operators	operator	NOUN
ejde-942	48	23	.	.	PUNCT
ejde-942	49	1	yang	yang	PROPN
ejde-942	49	2	et	et	PROPN
ejde-942	49	3	al	al	PROPN
ejde-942	49	4	.	.	PUNCT
ejde-942	50	1	[	[	X
ejde-942	50	2	20	20	NUM
ejde-942	50	3	]	]	PUNCT
ejde-942	50	4	further	far	ADV
ejde-942	50	5	advanced	advance	VERB
ejde-942	50	6	the	the	DET
ejde-942	50	7	field	field	NOUN
ejde-942	50	8	by	by	ADP
ejde-942	50	9	developing	develop	VERB
ejde-942	50	10	a	a	DET
ejde-942	50	11	novel	novel	ADJ
ejde-942	50	12	class	class	NOUN
ejde-942	50	13	of	of	ADP
ejde-942	50	14	physics	physics	NOUN
ejde-942	50	15	-	-	PUNCT
ejde-942	50	16	informed	inform	VERB
ejde-942	50	17	generative	generative	ADJ
ejde-942	50	18	adversarial	adversarial	ADJ
ejde-942	50	19	networks	network	NOUN
ejde-942	50	20	,	,	PUNCT
ejde-942	50	21	which	which	PRON
ejde-942	50	22	effectively	effectively	ADV
ejde-942	50	23	tackle	tackle	VERB
ejde-942	50	24	forward	forward	ADV
ejde-942	50	25	,	,	PUNCT
ejde-942	50	26	inverse	inverse	NOUN
ejde-942	50	27	,	,	PUNCT
ejde-942	50	28	and	and	CCONJ
ejde-942	50	29	mixed	mixed	ADJ
ejde-942	50	30	stochastic	stochastic	ADJ
ejde-942	50	31	problems	problem	NOUN
ejde-942	50	32	in	in	ADP
ejde-942	50	33	a	a	DET
ejde-942	50	34	unified	unified	ADJ
ejde-942	50	35	manner	manner	NOUN
ejde-942	50	36	,	,	PUNCT
ejde-942	50	37	relying	rely	VERB
ejde-942	50	38	on	on	ADP
ejde-942	50	39	a	a	DET
ejde-942	50	40	limited	limited	ADJ
ejde-942	50	41	number	number	NOUN
ejde-942	50	42	of	of	ADP
ejde-942	50	43	scattered	scatter	VERB
ejde-942	50	44	measurements	measurement	NOUN
ejde-942	50	45	.	.	PUNCT
ejde-942	51	1	lu	lu	VERB
ejde-942	51	2	et	et	PROPN
ejde-942	51	3	al	al	PROPN
ejde-942	51	4	.	.	PUNCT
ejde-942	52	1	[	[	X
ejde-942	52	2	16	16	NUM
ejde-942	52	3	]	]	PUNCT
ejde-942	52	4	have	have	AUX
ejde-942	52	5	presented	present	VERB
ejde-942	52	6	a	a	DET
ejde-942	52	7	python	python	NOUN
ejde-942	52	8	library	library	NOUN
ejde-942	52	9	,	,	PUNCT
ejde-942	52	10	named	name	VERB
ejde-942	52	11	deepxde	deepxde	NOUN
ejde-942	52	12	,	,	PUNCT
ejde-942	52	13	specifically	specifically	ADV
ejde-942	52	14	designed	design	VERB
ejde-942	52	15	for	for	ADP
ejde-942	52	16	pinns	pinn	NOUN
ejde-942	52	17	.	.	PUNCT
ejde-942	53	1	this	this	DET
ejde-942	53	2	library	library	NOUN
ejde-942	53	3	serves	serve	VERB
ejde-942	53	4	dual	dual	ADJ
ejde-942	53	5	purposes	purpose	NOUN
ejde-942	53	6	:	:	PUNCT
ejde-942	53	7	as	as	ADP
ejde-942	53	8	an	an	DET
ejde-942	53	9	educational	educational	ADJ
ejde-942	53	10	tool	tool	NOUN
ejde-942	53	11	for	for	ADP
ejde-942	53	12	classroom	classroom	NOUN
ejde-942	53	13	use	use	NOUN
ejde-942	53	14	and	and	CCONJ
ejde-942	53	15	as	as	ADP
ejde-942	53	16	a	a	DET
ejde-942	53	17	research	research	NOUN
ejde-942	53	18	tool	tool	NOUN
ejde-942	53	19	for	for	ADP
ejde-942	53	20	addressing	address	VERB
ejde-942	53	21	computational	computational	ADJ
ejde-942	53	22	science	science	NOUN
ejde-942	53	23	and	and	CCONJ
ejde-942	53	24	engineering	engineering	NOUN
ejde-942	53	25	challenges	challenge	NOUN
ejde-942	53	26	.	.	PUNCT
ejde-942	54	1	for	for	ADP
ejde-942	54	2	specific	specific	ADJ
ejde-942	54	3	mathematical	mathematical	ADJ
ejde-942	54	4	problems	problem	NOUN
ejde-942	54	5	,	,	PUNCT
ejde-942	54	6	such	such	ADJ
ejde-942	54	7	as	as	ADP
ejde-942	54	8	identifying	identify	VERB
ejde-942	54	9	hamiltonian	hamiltonian	ADJ
ejde-942	54	10	systems	system	NOUN
ejde-942	54	11	,	,	PUNCT
ejde-942	54	12	mattheakis	mattheakis	PROPN
ejde-942	54	13	et	et	PROPN
ejde-942	54	14	al	al	PROPN
ejde-942	54	15	.	.	PUNCT
ejde-942	55	1	[	[	X
ejde-942	55	2	7	7	NUM
ejde-942	55	3	]	]	PUNCT
ejde-942	55	4	developed	develop	VERB
ejde-942	55	5	a	a	DET
ejde-942	55	6	hamiltonian	hamiltonian	ADJ
ejde-942	55	7	neural	neural	ADJ
ejde-942	55	8	network	network	NOUN
ejde-942	55	9	(	(	PUNCT
ejde-942	55	10	hnn	hnn	PROPN
ejde-942	55	11	)	)	PUNCT
ejde-942	55	12	with	with	ADP
ejde-942	55	13	the	the	DET
ejde-942	55	14	primary	primary	ADJ
ejde-942	55	15	aim	aim	NOUN
ejde-942	55	16	of	of	ADP
ejde-942	55	17	conserving	conserve	VERB
ejde-942	55	18	the	the	DET
ejde-942	55	19	symplectic	symplectic	ADJ
ejde-942	55	20	structure	structure	NOUN
ejde-942	55	21	of	of	ADP
ejde-942	55	22	the	the	DET
ejde-942	55	23	hamiltonian	hamiltonian	ADJ
ejde-942	55	24	system	system	NOUN
ejde-942	55	25	.	.	PUNCT
ejde-942	56	1	this	this	PRON
ejde-942	56	2	allows	allow	VERB
ejde-942	56	3	for	for	ADP
ejde-942	56	4	more	more	ADV
ejde-942	56	5	accurate	accurate	ADJ
ejde-942	56	6	learning	learning	NOUN
ejde-942	56	7	and	and	CCONJ
ejde-942	56	8	prediction	prediction	NOUN
ejde-942	56	9	of	of	ADP
ejde-942	56	10	the	the	DET
ejde-942	56	11	dynamic	dynamic	ADJ
ejde-942	56	12	behavior	behavior	NOUN
ejde-942	56	13	of	of	ADP
ejde-942	56	14	the	the	DET
ejde-942	56	15	ejde-2024/53	ejde-2024/53	ADJ
ejde-942	56	16	deep	deep	ADJ
ejde-942	56	17	learning	learning	NOUN
ejde-942	56	18	method	method	NOUN
ejde-942	56	19	for	for	ADP
ejde-942	56	20	finding	find	VERB
ejde-942	56	21	eigenpairs	eigenpair	NOUN
ejde-942	56	22	3	3	NUM
ejde-942	56	23	system	system	NOUN
ejde-942	56	24	.	.	PUNCT
ejde-942	57	1	jin	jin	NOUN
ejde-942	57	2	et	et	PROPN
ejde-942	57	3	al	al	PROPN
ejde-942	57	4	.	.	PUNCT
ejde-942	58	1	[	[	X
ejde-942	58	2	13	13	NUM
ejde-942	58	3	]	]	PUNCT
ejde-942	58	4	have	have	AUX
ejde-942	58	5	developed	develop	VERB
ejde-942	58	6	symplectic	symplectic	ADJ
ejde-942	58	7	networks	network	NOUN
ejde-942	58	8	(	(	PUNCT
ejde-942	58	9	sympnets	sympnet	NOUN
ejde-942	58	10	)	)	PUNCT
ejde-942	58	11	to	to	PART
ejde-942	58	12	effectively	effectively	ADV
ejde-942	58	13	learn	learn	VERB
ejde-942	58	14	the	the	DET
ejde-942	58	15	symplectic	symplectic	ADJ
ejde-942	58	16	flow	flow	NOUN
ejde-942	58	17	within	within	ADP
ejde-942	58	18	hamiltonian	hamiltonian	ADJ
ejde-942	58	19	systems	system	NOUN
ejde-942	58	20	.	.	PUNCT
ejde-942	59	1	moreover	moreover	ADV
ejde-942	59	2	,	,	PUNCT
ejde-942	59	3	they	they	PRON
ejde-942	59	4	[	[	X
ejde-942	59	5	12	12	NUM
ejde-942	59	6	]	]	PUNCT
ejde-942	59	7	have	have	AUX
ejde-942	59	8	also	also	ADV
ejde-942	59	9	proposed	propose	VERB
ejde-942	59	10	poisson	poisson	PROPN
ejde-942	59	11	neural	neural	ADJ
ejde-942	59	12	networks	network	NOUN
ejde-942	59	13	(	(	PUNCT
ejde-942	59	14	pnns	pnn	NOUN
ejde-942	59	15	)	)	PUNCT
ejde-942	59	16	which	which	PRON
ejde-942	59	17	are	be	AUX
ejde-942	59	18	specifically	specifically	ADV
ejde-942	59	19	designed	design	VERB
ejde-942	59	20	to	to	PART
ejde-942	59	21	learn	learn	VERB
ejde-942	59	22	poisson	poisson	NOUN
ejde-942	59	23	systems	system	NOUN
ejde-942	59	24	and	and	CCONJ
ejde-942	59	25	accurately	accurately	ADV
ejde-942	59	26	capture	capture	VERB
ejde-942	59	27	the	the	DET
ejde-942	59	28	trajectories	trajectory	NOUN
ejde-942	59	29	of	of	ADP
ejde-942	59	30	autonomous	autonomous	ADJ
ejde-942	59	31	systems	system	NOUN
ejde-942	59	32	,	,	PUNCT
ejde-942	59	33	all	all	PRON
ejde-942	59	34	derived	derive	VERB
ejde-942	59	35	directly	directly	ADV
ejde-942	59	36	from	from	ADP
ejde-942	59	37	data	datum	NOUN
ejde-942	59	38	.	.	PUNCT
ejde-942	60	1	lu	lu	PROPN
ejde-942	60	2	et	et	PROPN
ejde-942	60	3	al	al	PROPN
ejde-942	60	4	.	.	PUNCT
ejde-942	61	1	[	[	X
ejde-942	61	2	15	15	NUM
ejde-942	61	3	]	]	PUNCT
ejde-942	61	4	proposed	propose	VERB
ejde-942	61	5	deeponet	deeponet	NOUN
ejde-942	61	6	,	,	PUNCT
ejde-942	61	7	which	which	PRON
ejde-942	61	8	has	have	AUX
ejde-942	61	9	demonstrated	demonstrate	VERB
ejde-942	61	10	its	its	PRON
ejde-942	61	11	efficacy	efficacy	NOUN
ejde-942	61	12	as	as	ADP
ejde-942	61	13	a	a	DET
ejde-942	61	14	powerful	powerful	ADJ
ejde-942	61	15	tool	tool	NOUN
ejde-942	61	16	for	for	ADP
ejde-942	61	17	handling	handle	VERB
ejde-942	61	18	nonlinear	nonlinear	ADJ
ejde-942	61	19	operators	operator	NOUN
ejde-942	61	20	using	use	VERB
ejde-942	61	21	supervised	supervised	ADJ
ejde-942	61	22	data	data	NOUN
ejde-942	61	23	-	-	PUNCT
ejde-942	61	24	driven	drive	VERB
ejde-942	61	25	methods	method	NOUN
ejde-942	61	26	.	.	PUNCT
ejde-942	62	1	what	what	PRON
ejde-942	62	2	’s	’	VERB
ejde-942	62	3	even	even	ADV
ejde-942	62	4	more	more	ADV
ejde-942	62	5	thrilling	thrilling	ADJ
ejde-942	62	6	is	be	AUX
ejde-942	62	7	the	the	DET
ejde-942	62	8	potential	potential	NOUN
ejde-942	62	9	that	that	PRON
ejde-942	62	10	emerges	emerge	VERB
ejde-942	62	11	from	from	ADP
ejde-942	62	12	merging	merge	VERB
ejde-942	62	13	deeponet	deeponet	NOUN
ejde-942	62	14	with	with	ADP
ejde-942	62	15	the	the	DET
ejde-942	62	16	physics	physics	NOUN
ejde-942	62	17	encoded	encode	VERB
ejde-942	62	18	by	by	ADP
ejde-942	62	19	pinns	pinn	NOUN
ejde-942	62	20	.	.	PUNCT
ejde-942	63	1	this	this	DET
ejde-942	63	2	union	union	NOUN
ejde-942	63	3	opens	open	VERB
ejde-942	63	4	up	up	ADP
ejde-942	63	5	the	the	DET
ejde-942	63	6	possibility	possibility	NOUN
ejde-942	63	7	of	of	ADP
ejde-942	63	8	achieving	achieve	VERB
ejde-942	63	9	precise	precise	ADJ
ejde-942	63	10	,	,	PUNCT
ejde-942	63	11	real	real	ADJ
ejde-942	63	12	-	-	PUNCT
ejde-942	63	13	time	time	NOUN
ejde-942	63	14	predictions	prediction	NOUN
ejde-942	63	15	in	in	ADP
ejde-942	63	16	many	many	ADJ
ejde-942	63	17	fields	field	NOUN
ejde-942	63	18	with	with	ADP
ejde-942	63	19	extrapolation	extrapolation	NOUN
ejde-942	63	20	capabilities	capability	NOUN
ejde-942	63	21	.	.	PUNCT
ejde-942	64	1	zang	zang	PROPN
ejde-942	64	2	et	et	PROPN
ejde-942	64	3	al	al	PROPN
ejde-942	64	4	.	.	PUNCT
ejde-942	65	1	[	[	X
ejde-942	65	2	21	21	NUM
ejde-942	65	3	]	]	PUNCT
ejde-942	65	4	presented	present	VERB
ejde-942	65	5	an	an	DET
ejde-942	65	6	adversarial	adversarial	ADJ
ejde-942	65	7	neural	neural	ADJ
ejde-942	65	8	network	network	NOUN
ejde-942	65	9	rooted	root	VERB
ejde-942	65	10	in	in	ADP
ejde-942	65	11	partial	partial	ADJ
ejde-942	65	12	differential	differential	ADJ
ejde-942	65	13	equation	equation	NOUN
ejde-942	65	14	weak	weak	ADJ
ejde-942	65	15	solutions	solution	NOUN
ejde-942	65	16	,	,	PUNCT
ejde-942	65	17	offering	offer	VERB
ejde-942	65	18	a	a	DET
ejde-942	65	19	pathway	pathway	NOUN
ejde-942	65	20	for	for	ADP
ejde-942	65	21	tackling	tackle	VERB
ejde-942	65	22	high	high	ADJ
ejde-942	65	23	-	-	PUNCT
ejde-942	65	24	dimensional	dimensional	ADJ
ejde-942	65	25	partial	partial	ADJ
ejde-942	65	26	differential	differential	NOUN
ejde-942	65	27	equations	equation	NOUN
ejde-942	65	28	.	.	PUNCT
ejde-942	66	1	for	for	ADP
ejde-942	66	2	further	further	ADJ
ejde-942	66	3	advancements	advancement	NOUN
ejde-942	66	4	related	relate	VERB
ejde-942	66	5	to	to	ADP
ejde-942	66	6	physics	physics	NOUN
ejde-942	66	7	-	-	PUNCT
ejde-942	66	8	informed	inform	VERB
ejde-942	66	9	machine	machine	NOUN
ejde-942	66	10	learning	learning	NOUN
ejde-942	66	11	,	,	PUNCT
ejde-942	66	12	we	we	PRON
ejde-942	66	13	highly	highly	ADV
ejde-942	66	14	recommend	recommend	VERB
ejde-942	66	15	delving	delve	VERB
ejde-942	66	16	into	into	ADP
ejde-942	66	17	the	the	DET
ejde-942	66	18	comprehensive	comprehensive	ADJ
ejde-942	66	19	review	review	NOUN
ejde-942	66	20	article	article	NOUN
ejde-942	66	21	[	[	X
ejde-942	66	22	14	14	NUM
ejde-942	66	23	]	]	PUNCT
ejde-942	66	24	for	for	ADP
ejde-942	66	25	a	a	DET
ejde-942	66	26	deeper	deep	ADJ
ejde-942	66	27	understanding	understanding	NOUN
ejde-942	66	28	.	.	PUNCT
ejde-942	67	1	the	the	DET
ejde-942	67	2	application	application	NOUN
ejde-942	67	3	of	of	ADP
ejde-942	67	4	deep	deep	ADJ
ejde-942	67	5	learning	learning	NOUN
ejde-942	67	6	techniques	technique	NOUN
ejde-942	67	7	in	in	ADP
ejde-942	67	8	addressing	address	VERB
ejde-942	67	9	eigenvalue	eigenvalue	NOUN
ejde-942	67	10	problems	problem	NOUN
ejde-942	67	11	has	have	AUX
ejde-942	67	12	witnessed	witness	VERB
ejde-942	67	13	significant	significant	ADJ
ejde-942	67	14	advancements	advancement	NOUN
ejde-942	67	15	.	.	PUNCT
ejde-942	68	1	pinn	pinn	PROPN
ejde-942	68	2	,	,	PUNCT
ejde-942	68	3	owing	owe	VERB
ejde-942	68	4	to	to	ADP
ejde-942	68	5	its	its	PRON
ejde-942	68	6	ease	ease	NOUN
ejde-942	68	7	of	of	ADP
ejde-942	68	8	implementation	implementation	NOUN
ejde-942	68	9	and	and	CCONJ
ejde-942	68	10	ability	ability	NOUN
ejde-942	68	11	to	to	PART
ejde-942	68	12	maintain	maintain	VERB
ejde-942	68	13	the	the	DET
ejde-942	68	14	self	self	NOUN
ejde-942	68	15	-	-	PUNCT
ejde-942	68	16	adjointness	adjointness	NOUN
ejde-942	68	17	of	of	ADP
ejde-942	68	18	the	the	DET
ejde-942	68	19	operator	operator	NOUN
ejde-942	68	20	throughout	throughout	ADP
ejde-942	68	21	the	the	DET
ejde-942	68	22	computational	computational	ADJ
ejde-942	68	23	process	process	NOUN
ejde-942	68	24	,	,	PUNCT
ejde-942	68	25	has	have	AUX
ejde-942	68	26	been	be	AUX
ejde-942	68	27	employed	employ	VERB
ejde-942	68	28	by	by	ADP
ejde-942	68	29	scholars	scholar	NOUN
ejde-942	68	30	to	to	PART
ejde-942	68	31	identify	identify	VERB
ejde-942	68	32	orthogonal	orthogonal	ADJ
ejde-942	68	33	eigenpairs	eigenpair	NOUN
ejde-942	68	34	.	.	PUNCT
ejde-942	69	1	jin	jin	PROPN
ejde-942	69	2	et	et	PROPN
ejde-942	69	3	al	al	PROPN
ejde-942	69	4	.	.	PUNCT
ejde-942	70	1	[	[	X
ejde-942	70	2	9	9	NUM
ejde-942	70	3	]	]	PUNCT
ejde-942	70	4	used	use	VERB
ejde-942	70	5	pinn	pinn	NOUN
ejde-942	70	6	to	to	PART
ejde-942	70	7	solve	solve	VERB
ejde-942	70	8	the	the	DET
ejde-942	70	9	quantum	quantum	ADJ
ejde-942	70	10	problems	problem	NOUN
ejde-942	70	11	related	relate	VERB
ejde-942	70	12	to	to	PART
ejde-942	70	13	finite	finite	VERB
ejde-942	70	14	wells	well	NOUN
ejde-942	70	15	,	,	PUNCT
ejde-942	70	16	multiple	multiple	ADJ
ejde-942	70	17	finite	finite	ADJ
ejde-942	70	18	wells	well	NOUN
ejde-942	70	19	,	,	PUNCT
ejde-942	70	20	and	and	CCONJ
ejde-942	70	21	hydrogen	hydrogen	NOUN
ejde-942	70	22	atom	atom	NOUN
ejde-942	70	23	eigenvalues	eigenvalue	NOUN
ejde-942	70	24	,	,	PUNCT
ejde-942	70	25	and	and	CCONJ
ejde-942	70	26	introduced	introduce	VERB
ejde-942	70	27	a	a	DET
ejde-942	70	28	strong	strong	ADJ
ejde-942	70	29	boundary	boundary	ADJ
ejde-942	70	30	value	value	NOUN
ejde-942	70	31	condition	condition	NOUN
ejde-942	70	32	.	.	PUNCT
ejde-942	71	1	however	however	ADV
ejde-942	71	2	,	,	PUNCT
ejde-942	71	3	their	their	PRON
ejde-942	71	4	investigation	investigation	NOUN
ejde-942	71	5	was	be	AUX
ejde-942	71	6	limited	limit	VERB
ejde-942	71	7	to	to	ADP
ejde-942	71	8	one	one	NUM
ejde-942	71	9	-	-	PUNCT
ejde-942	71	10	dimensional	dimensional	ADJ
ejde-942	71	11	homogeneous	homogeneous	ADJ
ejde-942	71	12	media	medium	NOUN
ejde-942	71	13	under	under	ADP
ejde-942	71	14	the	the	DET
ejde-942	71	15	dirichlet	dirichlet	PROPN
ejde-942	71	16	boundary	boundary	ADJ
ejde-942	71	17	value	value	NOUN
ejde-942	71	18	condition	condition	NOUN
ejde-942	71	19	.	.	PUNCT
ejde-942	72	1	holliday	holliday	PROPN
ejde-942	72	2	et	et	PROPN
ejde-942	72	3	al	al	PROPN
ejde-942	72	4	.	.	PUNCT
ejde-942	73	1	[	[	X
ejde-942	73	2	8	8	NUM
ejde-942	73	3	]	]	PUNCT
ejde-942	73	4	extended	extended	ADJ
ejde-942	73	5	jin	jin	NOUN
ejde-942	73	6	’s	’s	PART
ejde-942	73	7	algorithm	algorithm	NOUN
ejde-942	73	8	to	to	ADP
ejde-942	73	9	quantum	quantum	PROPN
ejde-942	73	10	billiard	billiard	NOUN
ejde-942	73	11	eigenvalue	eigenvalue	NOUN
ejde-942	73	12	problems	problem	NOUN
ejde-942	73	13	,	,	PUNCT
ejde-942	73	14	successfully	successfully	ADV
ejde-942	73	15	identifying	identify	VERB
ejde-942	73	16	the	the	DET
ejde-942	73	17	eigenvalues	eigenvalue	NOUN
ejde-942	73	18	and	and	CCONJ
ejde-942	73	19	eigenfunctions	eigenfunction	NOUN
ejde-942	73	20	of	of	ADP
ejde-942	73	21	the	the	DET
ejde-942	73	22	stationary	stationary	ADJ
ejde-942	73	23	schrödinger	schrödinger	NOUN
ejde-942	73	24	differential	differential	ADJ
ejde-942	73	25	equation	equation	NOUN
ejde-942	73	26	with	with	ADP
ejde-942	73	27	dirichlet	dirichlet	PROPN
ejde-942	73	28	boundary	boundary	PROPN
ejde-942	73	29	value	value	NOUN
ejde-942	73	30	conditions	condition	NOUN
ejde-942	73	31	in	in	ADP
ejde-942	73	32	two	two	NUM
ejde-942	73	33	-	-	PUNCT
ejde-942	73	34	dimensional	dimensional	ADJ
ejde-942	73	35	domains	domain	NOUN
ejde-942	73	36	.	.	PUNCT
ejde-942	74	1	ben	ben	PROPN
ejde-942	74	2	-	-	PUNCT
ejde-942	74	3	shuaul	shuaul	PROPN
ejde-942	74	4	et	et	PROPN
ejde-942	74	5	al	al	PROPN
ejde-942	74	6	.	.	PUNCT
ejde-942	75	1	[	[	X
ejde-942	75	2	5	5	NUM
ejde-942	75	3	]	]	PUNCT
ejde-942	75	4	have	have	AUX
ejde-942	75	5	effectively	effectively	ADV
ejde-942	75	6	employed	employ	VERB
ejde-942	75	7	pinn	pinn	NOUN
ejde-942	75	8	techniques	technique	NOUN
ejde-942	75	9	to	to	PART
ejde-942	75	10	tackle	tackle	VERB
ejde-942	75	11	the	the	DET
ejde-942	75	12	problem	problem	NOUN
ejde-942	75	13	of	of	ADP
ejde-942	75	14	finding	find	VERB
ejde-942	75	15	the	the	DET
ejde-942	75	16	smallest	small	ADJ
ejde-942	75	17	eigenpairs	eigenpair	NOUN
ejde-942	75	18	for	for	ADP
ejde-942	75	19	one	one	NUM
ejde-942	75	20	-	-	PUNCT
ejde-942	75	21	dimensional	dimensional	ADJ
ejde-942	75	22	sturm	sturm	NOUN
ejde-942	75	23	-	-	PUNCT
ejde-942	75	24	liouville	liouville	VERB
ejde-942	75	25	boundary	boundary	ADJ
ejde-942	75	26	value	value	NOUN
ejde-942	75	27	problems	problem	NOUN
ejde-942	75	28	.	.	PUNCT
ejde-942	76	1	the	the	DET
ejde-942	76	2	strength	strength	NOUN
ejde-942	76	3	of	of	ADP
ejde-942	76	4	their	their	PRON
ejde-942	76	5	approach	approach	NOUN
ejde-942	76	6	lies	lie	VERB
ejde-942	76	7	in	in	ADP
ejde-942	76	8	its	its	PRON
ejde-942	76	9	ability	ability	NOUN
ejde-942	76	10	to	to	PART
ejde-942	76	11	simultaneously	simultaneously	ADV
ejde-942	76	12	identify	identify	VERB
ejde-942	76	13	multiple	multiple	ADJ
ejde-942	76	14	eigenpairs	eigenpair	NOUN
ejde-942	76	15	.	.	PUNCT
ejde-942	77	1	however	however	ADV
ejde-942	77	2	,	,	PUNCT
ejde-942	77	3	their	their	PRON
ejde-942	77	4	method	method	NOUN
ejde-942	77	5	,	,	PUNCT
ejde-942	77	6	which	which	PRON
ejde-942	77	7	incorporates	incorporate	VERB
ejde-942	77	8	the	the	DET
ejde-942	77	9	rayleigh	rayleigh	PROPN
ejde-942	77	10	quotient	quotient	NOUN
ejde-942	77	11	for	for	ADP
ejde-942	77	12	eigenvalue	eigenvalue	ADJ
ejde-942	77	13	calculation	calculation	NOUN
ejde-942	77	14	,	,	PUNCT
ejde-942	77	15	can	can	AUX
ejde-942	77	16	be	be	AUX
ejde-942	77	17	challenging	challenge	VERB
ejde-942	77	18	for	for	ADP
ejde-942	77	19	non	non	NOUN
ejde-942	77	20	-	-	NOUN
ejde-942	77	21	experts	expert	NOUN
ejde-942	77	22	and	and	CCONJ
ejde-942	77	23	may	may	AUX
ejde-942	77	24	yield	yield	VERB
ejde-942	77	25	lower	low	ADJ
ejde-942	77	26	precision	precision	NOUN
ejde-942	77	27	results	result	NOUN
ejde-942	77	28	.	.	PUNCT
ejde-942	78	1	additionally	additionally	ADV
ejde-942	78	2	,	,	PUNCT
ejde-942	78	3	their	their	PRON
ejde-942	78	4	focus	focus	NOUN
ejde-942	78	5	is	be	AUX
ejde-942	78	6	specifically	specifically	ADV
ejde-942	78	7	on	on	ADP
ejde-942	78	8	scenarios	scenario	NOUN
ejde-942	78	9	involving	involve	VERB
ejde-942	78	10	the	the	DET
ejde-942	78	11	laplace	laplace	NOUN
ejde-942	78	12	operator	operator	NOUN
ejde-942	78	13	as	as	ADP
ejde-942	78	14	the	the	DET
ejde-942	78	15	differential	differential	NOUN
ejde-942	78	16	operator	operator	NOUN
ejde-942	78	17	,	,	PUNCT
ejde-942	78	18	disregarding	disregard	VERB
ejde-942	78	19	inhomogeneous	inhomogeneous	ADJ
ejde-942	78	20	cases	case	NOUN
ejde-942	78	21	.	.	PUNCT
ejde-942	79	1	furthermore	furthermore	ADV
ejde-942	79	2	,	,	PUNCT
ejde-942	79	3	their	their	PRON
ejde-942	79	4	framework	framework	NOUN
ejde-942	79	5	is	be	AUX
ejde-942	79	6	limited	limit	VERB
ejde-942	79	7	to	to	ADP
ejde-942	79	8	dirichlet	dirichlet	PROPN
ejde-942	79	9	boundary	boundary	ADJ
ejde-942	79	10	value	value	NOUN
ejde-942	79	11	condition	condition	NOUN
ejde-942	79	12	and	and	CCONJ
ejde-942	79	13	does	do	AUX
ejde-942	79	14	not	not	PART
ejde-942	79	15	account	account	VERB
ejde-942	79	16	for	for	ADP
ejde-942	79	17	derivative	derivative	ADJ
ejde-942	79	18	boundary	boundary	ADJ
ejde-942	79	19	situations	situation	NOUN
ejde-942	79	20	.	.	PUNCT
ejde-942	80	1	unlike	unlike	ADP
ejde-942	80	2	the	the	DET
ejde-942	80	3	dirichlet	dirichlet	PROPN
ejde-942	80	4	boundary	boundary	NOUN
ejde-942	80	5	,	,	PUNCT
ejde-942	80	6	the	the	DET
ejde-942	80	7	discrete	discrete	ADJ
ejde-942	80	8	matrix	matrix	NOUN
ejde-942	80	9	corresponding	correspond	VERB
ejde-942	80	10	to	to	ADP
ejde-942	80	11	the	the	DET
ejde-942	80	12	sturm	sturm	NOUN
ejde-942	80	13	-	-	PUNCT
ejde-942	80	14	liouville	liouville	VERB
ejde-942	80	15	differential	differential	NOUN
ejde-942	80	16	operator	operator	NOUN
ejde-942	80	17	under	under	ADP
ejde-942	80	18	derivative	derivative	ADJ
ejde-942	80	19	boundaries	boundary	NOUN
ejde-942	80	20	is	be	AUX
ejde-942	80	21	not	not	PART
ejde-942	80	22	self	self	NOUN
ejde-942	80	23	-	-	PUNCT
ejde-942	80	24	adjoint	adjoint	NOUN
ejde-942	80	25	when	when	SCONJ
ejde-942	80	26	using	use	VERB
ejde-942	80	27	classical	classical	ADJ
ejde-942	80	28	numerical	numerical	ADJ
ejde-942	80	29	methods	method	NOUN
ejde-942	80	30	.	.	PUNCT
ejde-942	81	1	this	this	DET
ejde-942	81	2	aspect	aspect	NOUN
ejde-942	81	3	underscores	underscore	VERB
ejde-942	81	4	the	the	DET
ejde-942	81	5	advantages	advantage	NOUN
ejde-942	81	6	of	of	ADP
ejde-942	81	7	leveraging	leverage	VERB
ejde-942	81	8	neural	neural	ADJ
ejde-942	81	9	networks	network	NOUN
ejde-942	81	10	for	for	ADP
ejde-942	81	11	addressing	address	VERB
ejde-942	81	12	such	such	ADJ
ejde-942	81	13	problems	problem	NOUN
ejde-942	81	14	.	.	PUNCT
ejde-942	82	1	in	in	ADP
ejde-942	82	2	this	this	DET
ejde-942	82	3	article	article	NOUN
ejde-942	82	4	,	,	PUNCT
ejde-942	82	5	we	we	PRON
ejde-942	82	6	consider	consider	VERB
ejde-942	82	7	how	how	SCONJ
ejde-942	82	8	to	to	PART
ejde-942	82	9	find	find	VERB
ejde-942	82	10	the	the	DET
ejde-942	82	11	smallest	small	ADJ
ejde-942	82	12	eigenpairs	eigenpair	NOUN
ejde-942	82	13	using	use	VERB
ejde-942	82	14	deep	deep	ADJ
ejde-942	82	15	learning	learning	NOUN
ejde-942	82	16	methods	method	NOUN
ejde-942	82	17	.	.	PUNCT
ejde-942	83	1	we	we	PRON
ejde-942	83	2	introduce	introduce	VERB
ejde-942	83	3	eigenvalues	eigenvalue	NOUN
ejde-942	83	4	as	as	ADP
ejde-942	83	5	trainable	trainable	ADJ
ejde-942	83	6	parameters	parameter	NOUN
ejde-942	83	7	,	,	PUNCT
ejde-942	83	8	construct	construct	VERB
ejde-942	83	9	a	a	DET
ejde-942	83	10	novel	novel	ADJ
ejde-942	83	11	cost	cost	NOUN
ejde-942	83	12	function	function	NOUN
ejde-942	83	13	,	,	PUNCT
ejde-942	83	14	add	add	VERB
ejde-942	83	15	an	an	DET
ejde-942	83	16	adaptive	adaptive	ADJ
ejde-942	83	17	hyper	hyper	ADJ
ejde-942	83	18	-	-	ADJ
ejde-942	83	19	parameter	parameter	ADJ
ejde-942	83	20	tuning	tuning	NOUN
ejde-942	83	21	strategy	strategy	NOUN
ejde-942	83	22	,	,	PUNCT
ejde-942	83	23	and	and	CCONJ
ejde-942	83	24	enhance	enhance	VERB
ejde-942	83	25	the	the	DET
ejde-942	83	26	speed	speed	NOUN
ejde-942	83	27	and	and	CCONJ
ejde-942	83	28	accuracy	accuracy	NOUN
ejde-942	83	29	of	of	ADP
ejde-942	83	30	training	training	NOUN
ejde-942	83	31	by	by	ADP
ejde-942	83	32	sequentially	sequentially	ADJ
ejde-942	83	33	training	training	NOUN
ejde-942	83	34	eigenpairs	eigenpair	NOUN
ejde-942	83	35	.	.	PUNCT
ejde-942	84	1	this	this	DET
ejde-942	84	2	setup	setup	NOUN
ejde-942	84	3	allows	allow	VERB
ejde-942	84	4	us	we	PRON
ejde-942	84	5	to	to	PART
ejde-942	84	6	compute	compute	VERB
ejde-942	84	7	a	a	DET
ejde-942	84	8	more	more	ADV
ejde-942	84	9	extensive	extensive	ADJ
ejde-942	84	10	set	set	NOUN
ejde-942	84	11	of	of	ADP
ejde-942	84	12	smallest	small	ADJ
ejde-942	84	13	orthogonal	orthogonal	ADJ
ejde-942	84	14	eigenpairs	eigenpair	NOUN
ejde-942	84	15	.	.	PUNCT
ejde-942	85	1	by	by	ADP
ejde-942	85	2	utilizing	utilize	VERB
ejde-942	85	3	a	a	DET
ejde-942	85	4	weighted	weight	VERB
ejde-942	85	5	inner	inner	ADJ
ejde-942	85	6	product	product	NOUN
ejde-942	85	7	space	space	NOUN
ejde-942	85	8	and	and	CCONJ
ejde-942	85	9	adaptive	adaptive	ADJ
ejde-942	85	10	hyper	hyper	NOUN
ejde-942	85	11	-	-	NOUN
ejde-942	85	12	parameters	parameter	NOUN
ejde-942	85	13	,	,	PUNCT
ejde-942	85	14	we	we	PRON
ejde-942	85	15	tackle	tackle	VERB
ejde-942	85	16	the	the	DET
ejde-942	85	17	sturm	sturm	NOUN
ejde-942	85	18	-	-	PUNCT
ejde-942	85	19	liouville	liouville	NOUN
ejde-942	85	20	eigenvalue	eigenvalue	NOUN
ejde-942	85	21	problem	problem	NOUN
ejde-942	85	22	under	under	ADP
ejde-942	85	23	varying	vary	VERB
ejde-942	85	24	boundary	boundary	ADJ
ejde-942	85	25	value	value	NOUN
ejde-942	85	26	conditions	condition	NOUN
ejde-942	85	27	in	in	ADP
ejde-942	85	28	inhomogeneous	inhomogeneous	ADJ
ejde-942	85	29	media	medium	NOUN
ejde-942	85	30	.	.	PUNCT
ejde-942	86	1	the	the	DET
ejde-942	86	2	simplicity	simplicity	NOUN
ejde-942	86	3	,	,	PUNCT
ejde-942	86	4	accuracy	accuracy	NOUN
ejde-942	86	5	,	,	PUNCT
ejde-942	86	6	and	and	CCONJ
ejde-942	86	7	interpretability	interpretability	NOUN
ejde-942	86	8	of	of	ADP
ejde-942	86	9	our	our	PRON
ejde-942	86	10	method	method	NOUN
ejde-942	86	11	4	4	NUM
ejde-942	86	12	s.	s.	PROPN
ejde-942	86	13	zhang	zhang	PROPN
ejde-942	86	14	,	,	PUNCT
ejde-942	86	15	j.	j.	PROPN
ejde-942	86	16	zu	zu	PROPN
ejde-942	86	17	,	,	PUNCT
ejde-942	86	18	j.	j.	PROPN
ejde-942	86	19	zhang	zhang	PROPN
ejde-942	86	20	ejde-2024/53	ejde-2024/53	PROPN
ejde-942	86	21	significantly	significantly	ADV
ejde-942	86	22	broaden	broaden	VERB
ejde-942	86	23	the	the	DET
ejde-942	86	24	applicability	applicability	NOUN
ejde-942	86	25	of	of	ADP
ejde-942	86	26	the	the	DET
ejde-942	86	27	proposed	propose	VERB
ejde-942	86	28	approach	approach	NOUN
ejde-942	86	29	.	.	PUNCT
ejde-942	87	1	quantitative	quantitative	ADJ
ejde-942	87	2	estimation	estimation	NOUN
ejde-942	87	3	of	of	ADP
ejde-942	87	4	eigenpairs	eigenpair	NOUN
ejde-942	87	5	is	be	AUX
ejde-942	87	6	given	give	VERB
ejde-942	87	7	for	for	ADP
ejde-942	87	8	the	the	DET
ejde-942	87	9	sturm	sturm	NOUN
ejde-942	87	10	-	-	PUNCT
ejde-942	87	11	liouville	liouville	NOUN
ejde-942	87	12	eigenvalue	eigenvalue	NOUN
ejde-942	87	13	problems	problem	NOUN
ejde-942	87	14	.	.	PUNCT
ejde-942	88	1	furthermore	furthermore	ADV
ejde-942	88	2	,	,	PUNCT
ejde-942	88	3	we	we	PRON
ejde-942	88	4	extend	extend	VERB
ejde-942	88	5	the	the	DET
ejde-942	88	6	proposed	propose	VERB
ejde-942	88	7	method	method	NOUN
ejde-942	88	8	to	to	ADP
ejde-942	88	9	two	two	NUM
ejde-942	88	10	-	-	PUNCT
ejde-942	88	11	dimensional	dimensional	ADJ
ejde-942	88	12	cases	case	NOUN
ejde-942	88	13	,	,	PUNCT
ejde-942	88	14	periodic	periodic	ADJ
ejde-942	88	15	scenarios	scenario	NOUN
ejde-942	88	16	.	.	PUNCT
ejde-942	89	1	the	the	DET
ejde-942	89	2	organization	organization	NOUN
ejde-942	89	3	of	of	ADP
ejde-942	89	4	this	this	DET
ejde-942	89	5	paper	paper	NOUN
ejde-942	89	6	is	be	AUX
ejde-942	89	7	as	as	SCONJ
ejde-942	89	8	follows	follow	VERB
ejde-942	89	9	.	.	PUNCT
ejde-942	90	1	in	in	ADP
ejde-942	90	2	section	section	NOUN
ejde-942	90	3	2	2	NUM
ejde-942	90	4	,	,	PUNCT
ejde-942	90	5	we	we	PRON
ejde-942	90	6	introduce	introduce	VERB
ejde-942	90	7	the	the	DET
ejde-942	90	8	problem	problem	NOUN
ejde-942	90	9	model	model	NOUN
ejde-942	90	10	,	,	PUNCT
ejde-942	90	11	construct	construct	VERB
ejde-942	90	12	a	a	DET
ejde-942	90	13	novel	novel	ADJ
ejde-942	90	14	cost	cost	NOUN
ejde-942	90	15	function	function	NOUN
ejde-942	90	16	,	,	PUNCT
ejde-942	90	17	and	and	CCONJ
ejde-942	90	18	present	present	VERB
ejde-942	90	19	our	our	PRON
ejde-942	90	20	algorithm	algorithm	NOUN
ejde-942	90	21	.	.	PUNCT
ejde-942	91	1	in	in	ADP
ejde-942	91	2	section	section	NOUN
ejde-942	91	3	3	3	NUM
ejde-942	91	4	,	,	PUNCT
ejde-942	91	5	we	we	PRON
ejde-942	91	6	give	give	VERB
ejde-942	91	7	a	a	DET
ejde-942	91	8	quantitative	quantitative	ADJ
ejde-942	91	9	estimation	estimation	NOUN
ejde-942	91	10	of	of	ADP
ejde-942	91	11	eigenpairs	eigenpair	NOUN
ejde-942	91	12	.	.	PUNCT
ejde-942	92	1	in	in	ADP
ejde-942	92	2	section	section	NOUN
ejde-942	92	3	4	4	NUM
ejde-942	92	4	,	,	PUNCT
ejde-942	92	5	we	we	PRON
ejde-942	92	6	compare	compare	VERB
ejde-942	92	7	our	our	PRON
ejde-942	92	8	approach	approach	NOUN
ejde-942	92	9	with	with	ADP
ejde-942	92	10	previous	previous	ADJ
ejde-942	92	11	algorithms	algorithm	NOUN
ejde-942	92	12	,	,	PUNCT
ejde-942	92	13	including	include	VERB
ejde-942	92	14	state	state	NOUN
ejde-942	92	15	-	-	PUNCT
ejde-942	92	16	of	of	ADP
ejde-942	92	17	-	-	PUNCT
ejde-942	92	18	the	the	DET
ejde-942	92	19	-	-	PUNCT
ejde-942	92	20	art	art	NOUN
ejde-942	92	21	deep	deep	ADJ
ejde-942	92	22	learning	learning	NOUN
ejde-942	92	23	methods	method	NOUN
ejde-942	92	24	and	and	CCONJ
ejde-942	92	25	classical	classical	ADJ
ejde-942	92	26	numerical	numerical	ADJ
ejde-942	92	27	techniques	technique	NOUN
ejde-942	92	28	.	.	PUNCT
ejde-942	93	1	section	section	NOUN
ejde-942	93	2	5	5	NUM
ejde-942	93	3	presents	present	VERB
ejde-942	93	4	extensive	extensive	ADJ
ejde-942	93	5	numerical	numerical	ADJ
ejde-942	93	6	experiments	experiment	NOUN
ejde-942	93	7	that	that	PRON
ejde-942	93	8	demonstrate	demonstrate	VERB
ejde-942	93	9	the	the	DET
ejde-942	93	10	efficiency	efficiency	NOUN
ejde-942	93	11	and	and	CCONJ
ejde-942	93	12	accuracy	accuracy	NOUN
ejde-942	93	13	of	of	ADP
ejde-942	93	14	our	our	PRON
ejde-942	93	15	method	method	NOUN
ejde-942	93	16	in	in	ADP
ejde-942	93	17	addressing	address	VERB
ejde-942	93	18	the	the	DET
ejde-942	93	19	eigenvalue	eigenvalue	PROPN
ejde-942	93	20	problem	problem	NOUN
ejde-942	93	21	under	under	ADP
ejde-942	93	22	various	various	ADJ
ejde-942	93	23	boundary	boundary	ADJ
ejde-942	93	24	conditions	condition	NOUN
ejde-942	93	25	.	.	PUNCT
ejde-942	94	1	finally	finally	ADV
ejde-942	94	2	,	,	PUNCT
ejde-942	94	3	in	in	ADP
ejde-942	94	4	section	section	NOUN
ejde-942	94	5	6	6	NUM
ejde-942	94	6	,	,	PUNCT
ejde-942	94	7	we	we	PRON
ejde-942	94	8	conclude	conclude	VERB
ejde-942	94	9	the	the	DET
ejde-942	94	10	paper	paper	NOUN
ejde-942	94	11	by	by	ADP
ejde-942	94	12	summarizing	summarize	VERB
ejde-942	94	13	our	our	PRON
ejde-942	94	14	key	key	ADJ
ejde-942	94	15	findings	finding	NOUN
ejde-942	94	16	and	and	CCONJ
ejde-942	94	17	contributions	contribution	NOUN
ejde-942	94	18	.	.	PUNCT
ejde-942	95	1	2	2	X
ejde-942	95	2	.	.	X
ejde-942	95	3	model	model	NOUN
ejde-942	95	4	problem	problem	NOUN
ejde-942	95	5	in	in	ADP
ejde-942	95	6	this	this	DET
ejde-942	95	7	article	article	NOUN
ejde-942	95	8	,	,	PUNCT
ejde-942	95	9	we	we	PRON
ejde-942	95	10	aim	aim	VERB
ejde-942	95	11	to	to	PART
ejde-942	95	12	find	find	VERB
ejde-942	95	13	the	the	DET
ejde-942	95	14	m	m	NOUN
ejde-942	95	15	smallest	small	ADJ
ejde-942	95	16	eigenvalues	eigenvalue	NOUN
ejde-942	95	17	and	and	CCONJ
ejde-942	95	18	eigenfunctions	eigenfunction	NOUN
ejde-942	95	19	of	of	ADP
ejde-942	95	20	the	the	DET
ejde-942	95	21	general	general	ADJ
ejde-942	95	22	sturm	sturm	PROPN
ejde-942	95	23	-	-	PUNCT
ejde-942	95	24	liouville	liouville	NOUN
ejde-942	95	25	eigenvalue	eigenvalue	NOUN
ejde-942	95	26	problems	problem	NOUN
ejde-942	95	27	(	(	PUNCT
ejde-942	95	28	p(x)φ′(x))′	p(x)φ′(x))′	PROPN
ejde-942	95	29	+	+	CCONJ
ejde-942	95	30	λσ(x)φ(x	λσ(x)φ(x	PROPN
ejde-942	95	31	)	)	PUNCT
ejde-942	95	32	=	=	SYM
ejde-942	96	1	0	0	NUM
ejde-942	96	2	,	,	PUNCT
ejde-942	96	3	(	(	PUNCT
ejde-942	96	4	2.1	2.1	NUM
ejde-942	96	5	)	)	PUNCT
ejde-942	96	6	with	with	ADP
ejde-942	96	7	homogeneous	homogeneous	ADJ
ejde-942	96	8	boundary	boundary	ADJ
ejde-942	96	9	value	value	NOUN
ejde-942	96	10	conditions	condition	NOUN
ejde-942	96	11	a1φ	a1φ	ADJ
ejde-942	96	12	′(0)−	′(0)−	PUNCT
ejde-942	96	13	b1φ(0	b1φ(0	PROPN
ejde-942	96	14	)	)	PUNCT
ejde-942	96	15	=	=	SYM
ejde-942	97	1	0	0	NUM
ejde-942	97	2	,	,	PUNCT
ejde-942	97	3	a2φ	a2φ	ADP
ejde-942	97	4	′(π	′(π	NOUN
ejde-942	97	5	)	)	PUNCT
ejde-942	97	6	+	+	CCONJ
ejde-942	97	7	b2φ(π	b2φ(π	NOUN
ejde-942	97	8	)	)	PUNCT
ejde-942	97	9	=	=	SYM
ejde-942	98	1	0	0	NUM
ejde-942	98	2	,	,	PUNCT
ejde-942	98	3	a2i	a2i	PROPN
ejde-942	98	4	+	+	CCONJ
ejde-942	98	5	b2i	b2i	PROPN
ejde-942	98	6	̸=	̸=	PROPN
ejde-942	98	7	0	0	NUM
ejde-942	98	8	,	,	PUNCT
ejde-942	98	9	i	i	PRON
ejde-942	98	10	=	=	NOUN
ejde-942	98	11	1	1	NUM
ejde-942	98	12	,	,	PUNCT
ejde-942	98	13	2	2	NUM
ejde-942	98	14	,	,	PUNCT
ejde-942	98	15	(	(	PUNCT
ejde-942	98	16	2.2	2.2	NUM
ejde-942	98	17	)	)	PUNCT
ejde-942	98	18	where	where	SCONJ
ejde-942	98	19	p	p	PROPN
ejde-942	98	20	∈	∈	PROPN
ejde-942	98	21	c1((0	c1((0	NOUN
ejde-942	98	22	,	,	PUNCT
ejde-942	98	23	π),r+	π),r+	PROPN
ejde-942	98	24	)	)	PUNCT
ejde-942	98	25	and	and	CCONJ
ejde-942	98	26	σ	σ	PROPN
ejde-942	98	27	∈	∈	PROPN
ejde-942	98	28	c((0	c((0	PROPN
ejde-942	98	29	,	,	PUNCT
ejde-942	98	30	π),r+	π),r+	PROPN
ejde-942	98	31	)	)	PUNCT
ejde-942	98	32	,	,	PUNCT
ejde-942	98	33	λ	λ	PROPN
ejde-942	98	34	is	be	AUX
ejde-942	98	35	the	the	DET
ejde-942	98	36	eigenvalue	eigenvalue	PROPN
ejde-942	98	37	and	and	CCONJ
ejde-942	98	38	φ	φ	PROPN
ejde-942	98	39	∈	∈	PROPN
ejde-942	98	40	c2((0	c2((0	PROPN
ejde-942	98	41	,	,	PUNCT
ejde-942	98	42	π),r	π),r	PROPN
ejde-942	98	43	)	)	PUNCT
ejde-942	98	44	is	be	AUX
ejde-942	98	45	the	the	DET
ejde-942	98	46	corresponding	corresponding	ADJ
ejde-942	98	47	eigenfunction	eigenfunction	NOUN
ejde-942	98	48	.	.	PUNCT
ejde-942	99	1	in	in	ADP
ejde-942	99	2	some	some	DET
ejde-942	99	3	instances	instance	NOUN
ejde-942	99	4	,	,	PUNCT
ejde-942	99	5	p	p	NOUN
ejde-942	99	6	and	and	CCONJ
ejde-942	99	7	σ	σ	PROPN
ejde-942	99	8	are	be	AUX
ejde-942	99	9	elastic	elastic	ADJ
ejde-942	99	10	coefficent	coefficent	NOUN
ejde-942	99	11	and	and	CCONJ
ejde-942	99	12	rock	rock	NOUN
ejde-942	99	13	density	density	NOUN
ejde-942	99	14	coefficient	coefficient	NOUN
ejde-942	99	15	respectively	respectively	ADV
ejde-942	99	16	,	,	PUNCT
ejde-942	99	17	see	see	VERB
ejde-942	99	18	[	[	X
ejde-942	99	19	3	3	NUM
ejde-942	99	20	]	]	PUNCT
ejde-942	99	21	.	.	PUNCT
ejde-942	100	1	by	by	ADP
ejde-942	100	2	a	a	DET
ejde-942	100	3	change	change	NOUN
ejde-942	100	4	of	of	ADP
ejde-942	100	5	variable	variable	NOUN
ejde-942	100	6	x→	x→	PUNCT
ejde-942	100	7	z	z	AUX
ejde-942	100	8	given	give	VERB
ejde-942	100	9	by	by	ADP
ejde-942	100	10	z	z	PROPN
ejde-942	100	11	=	=	SYM
ejde-942	100	12	∫	∫	PROPN
ejde-942	100	13	x	x	SYM
ejde-942	100	14	0	0	NUM
ejde-942	100	15	√	√	PROPN
ejde-942	100	16	σ(s	σ(s	PROPN
ejde-942	100	17	)	)	PUNCT
ejde-942	100	18	p(s	p(s	PROPN
ejde-942	100	19	)	)	PUNCT
ejde-942	100	20	ds	ds	NOUN
ejde-942	100	21	,	,	PUNCT
ejde-942	100	22	equation	equation	NOUN
ejde-942	100	23	(	(	PUNCT
ejde-942	100	24	2.1	2.1	NUM
ejde-942	100	25	)	)	PUNCT
ejde-942	100	26	leads	lead	VERB
ejde-942	100	27	to	to	ADP
ejde-942	100	28	(	(	PUNCT
ejde-942	100	29	ρ(z)φ′(z))′	ρ(z)φ′(z))′	PROPN
ejde-942	100	30	+	+	CCONJ
ejde-942	101	1	λρ(z)φ(z	λρ(z)φ(z	NOUN
ejde-942	101	2	)	)	PUNCT
ejde-942	101	3	=	=	SYM
ejde-942	102	1	0	0	NUM
ejde-942	102	2	,	,	PUNCT
ejde-942	102	3	(	(	PUNCT
ejde-942	102	4	2.3	2.3	NUM
ejde-942	102	5	)	)	PUNCT
ejde-942	102	6	where	where	SCONJ
ejde-942	102	7	ρ(z	ρ(z	NOUN
ejde-942	102	8	)	)	PUNCT
ejde-942	102	9	=	=	SYM
ejde-942	102	10	√	√	NUM
ejde-942	102	11	σ(z)p(z	σ(z)p(z	NOUN
ejde-942	102	12	)	)	PUNCT
ejde-942	102	13	is	be	AUX
ejde-942	102	14	the	the	DET
ejde-942	102	15	impedance	impedance	NOUN
ejde-942	102	16	function	function	NOUN
ejde-942	102	17	.	.	PUNCT
ejde-942	103	1	the	the	DET
ejde-942	103	2	mathematical	mathematical	ADJ
ejde-942	103	3	theoretical	theoretical	ADJ
ejde-942	103	4	results	result	NOUN
ejde-942	103	5	for	for	ADP
ejde-942	103	6	the	the	DET
ejde-942	103	7	sturm	sturm	PROPN
ejde-942	103	8	-	-	PUNCT
ejde-942	103	9	liouville	liouville	NOUN
ejde-942	103	10	eigenvalue	eigenvalue	NOUN
ejde-942	103	11	problems	problem	NOUN
ejde-942	103	12	(	(	PUNCT
ejde-942	103	13	2.1	2.1	NUM
ejde-942	103	14	)	)	PUNCT
ejde-942	103	15	or	or	CCONJ
ejde-942	103	16	(	(	PUNCT
ejde-942	103	17	2.3	2.3	NUM
ejde-942	103	18	)	)	PUNCT
ejde-942	103	19	are	be	AUX
ejde-942	103	20	relatively	relatively	ADV
ejde-942	103	21	abundant	abundant	ADJ
ejde-942	103	22	.	.	PUNCT
ejde-942	104	1	let	let	VERB
ejde-942	104	2	ω	ω	NOUN
ejde-942	104	3	=	=	SYM
ejde-942	104	4	(	(	PUNCT
ejde-942	104	5	0	0	NUM
ejde-942	104	6	,	,	PUNCT
ejde-942	104	7	π	π	NOUN
ejde-942	104	8	)	)	PUNCT
ejde-942	104	9	.	.	PUNCT
ejde-942	105	1	define	define	VERB
ejde-942	105	2	the	the	DET
ejde-942	105	3	sturm	sturm	NOUN
ejde-942	105	4	-	-	PUNCT
ejde-942	105	5	liouville	liouville	NOUN
ejde-942	105	6	operator	operator	NOUN
ejde-942	105	7	in	in	ADP
ejde-942	105	8	inhomogeneous	inhomogeneous	ADJ
ejde-942	105	9	media	medium	NOUN
ejde-942	105	10	by	by	ADP
ejde-942	105	11	l[φ	l[φ	NOUN
ejde-942	105	12	]	]	PUNCT
ejde-942	105	13	:	:	PUNCT
ejde-942	106	1	=	=	SYM
ejde-942	106	2	−	−	PROPN
ejde-942	106	3	(	(	PUNCT
ejde-942	106	4	p(x)φ′(x))′	p(x)φ′(x))′	PROPN
ejde-942	106	5	σ(x	σ(x	PROPN
ejde-942	106	6	)	)	PUNCT
ejde-942	106	7	=	=	SYM
ejde-942	106	8	λφ	λφ	NOUN
ejde-942	106	9	,	,	PUNCT
ejde-942	106	10	in	in	ADP
ejde-942	106	11	ω	ω	NUM
ejde-942	106	12	.	.	PUNCT
ejde-942	106	13	(	(	PUNCT
ejde-942	106	14	2.4	2.4	NUM
ejde-942	106	15	)	)	PUNCT
ejde-942	106	16	define	define	VERB
ejde-942	106	17	the	the	DET
ejde-942	106	18	weighted	weight	VERB
ejde-942	106	19	inner	inner	ADJ
ejde-942	106	20	product	product	NOUN
ejde-942	106	21	by	by	ADP
ejde-942	106	22	(	(	PUNCT
ejde-942	106	23	φ	φ	NOUN
ejde-942	106	24	,	,	PUNCT
ejde-942	106	25	ψ)σ	ψ)σ	NOUN
ejde-942	106	26	:	:	PUNCT
ejde-942	106	27	=	=	SYM
ejde-942	106	28	∫	∫	PROPN
ejde-942	106	29	ω	ω	NUM
ejde-942	106	30	φψ∗σdx	φψ∗σdx	PROPN
ejde-942	106	31	,	,	PUNCT
ejde-942	106	32	φ	φ	X
ejde-942	106	33	,	,	PUNCT
ejde-942	106	34	ψ	ψ	X
ejde-942	106	35	∈	∈	PROPN
ejde-942	106	36	l2(ω	l2(ω	PROPN
ejde-942	106	37	,	,	PUNCT
ejde-942	106	38	σdx	σdx	PROPN
ejde-942	106	39	)	)	PUNCT
ejde-942	106	40	,	,	PUNCT
ejde-942	106	41	which	which	PRON
ejde-942	106	42	induces	induce	VERB
ejde-942	106	43	the	the	DET
ejde-942	106	44	l2	l2	NOUN
ejde-942	106	45	norm	norm	NOUN
ejde-942	107	1	∥	∥	X
ejde-942	107	2	·	·	PUNCT
ejde-942	107	3	∥σ	∥σ	NOUN
ejde-942	107	4	.	.	PUNCT
ejde-942	108	1	note	note	VERB
ejde-942	108	2	that	that	SCONJ
ejde-942	108	3	when	when	SCONJ
ejde-942	108	4	σ	σ	PROPN
ejde-942	108	5	≡	≡	PROPN
ejde-942	108	6	1	1	NUM
ejde-942	108	7	on	on	ADP
ejde-942	108	8	ω	ω	NUM
ejde-942	108	9	,	,	PUNCT
ejde-942	108	10	∥	∥	X
ejde-942	108	11	·	·	PUNCT
ejde-942	108	12	∥σ	∥σ	NOUN
ejde-942	108	13	reduces	reduce	VERB
ejde-942	108	14	to	to	PART
ejde-942	108	15	be	be	AUX
ejde-942	108	16	the	the	DET
ejde-942	108	17	standard	standard	ADJ
ejde-942	108	18	l2	l2	NOUN
ejde-942	108	19	norm	norm	NOUN
ejde-942	108	20	,	,	PUNCT
ejde-942	108	21	which	which	PRON
ejde-942	108	22	is	be	AUX
ejde-942	108	23	denoted	denote	VERB
ejde-942	108	24	by	by	ADP
ejde-942	108	25	∥	∥	NOUN
ejde-942	108	26	·	·	PUNCT
ejde-942	108	27	∥.	∥.	NOUN
ejde-942	109	1	it	it	PRON
ejde-942	109	2	is	be	AUX
ejde-942	109	3	well	well	ADV
ejde-942	109	4	known	know	VERB
ejde-942	109	5	that	that	SCONJ
ejde-942	109	6	l	l	NOUN
ejde-942	109	7	is	be	AUX
ejde-942	109	8	a	a	DET
ejde-942	109	9	selfadjoint	selfadjoint	NOUN
ejde-942	109	10	operator	operator	NOUN
ejde-942	109	11	,	,	PUNCT
ejde-942	109	12	the	the	DET
ejde-942	109	13	eigenvectors	eigenvector	NOUN
ejde-942	109	14	of	of	ADP
ejde-942	109	15	l	l	NOUN
ejde-942	109	16	construct	construct	VERB
ejde-942	109	17	an	an	DET
ejde-942	109	18	orthonormal	orthonormal	ADJ
ejde-942	109	19	basis	basis	NOUN
ejde-942	109	20	(	(	PUNCT
ejde-942	109	21	φk)k∈n+	φk)k∈n+	NOUN
ejde-942	109	22	of	of	ADP
ejde-942	109	23	l2((0	l2((0	PROPN
ejde-942	109	24	,	,	PUNCT
ejde-942	109	25	π	π	PROPN
ejde-942	109	26	)	)	PUNCT
ejde-942	109	27	,	,	PUNCT
ejde-942	109	28	σdx	σdx	PROPN
ejde-942	109	29	):	):	PUNCT
ejde-942	109	30	l[φk	l[φk	NOUN
ejde-942	109	31	]	]	X
ejde-942	109	32	=	=	SYM
ejde-942	109	33	λkφk	λkφk	NOUN
ejde-942	109	34	,	,	PUNCT
ejde-942	109	35	k	k	PROPN
ejde-942	109	36	=	=	SYM
ejde-942	109	37	1	1	NUM
ejde-942	109	38	,	,	PUNCT
ejde-942	109	39	2	2	NUM
ejde-942	109	40	,	,	PUNCT
ejde-942	109	41	.	.	PUNCT
ejde-942	109	42	.	.	PUNCT
ejde-942	109	43	.	.	PUNCT
ejde-942	110	1	,	,	PUNCT
ejde-942	110	2	(	(	PUNCT
ejde-942	110	3	2.5	2.5	NUM
ejde-942	110	4	)	)	PUNCT
ejde-942	110	5	where	where	SCONJ
ejde-942	110	6	(	(	PUNCT
ejde-942	110	7	λk)k∈n	λk)k∈n	PROPN
ejde-942	110	8	is	be	AUX
ejde-942	110	9	increasingly	increasingly	ADV
ejde-942	110	10	approach	approach	ADJ
ejde-942	110	11	to	to	ADP
ejde-942	110	12	+	+	NOUN
ejde-942	110	13	∞	∞	PROPN
ejde-942	110	14	as	as	ADP
ejde-942	110	15	k	k	PROPN
ejde-942	110	16	→	→	PROPN
ejde-942	110	17	+	+	PROPN
ejde-942	110	18	∞.	∞.	PROPN
ejde-942	110	19	finding	find	VERB
ejde-942	110	20	analytical	analytical	ADJ
ejde-942	110	21	expressions	expression	NOUN
ejde-942	110	22	for	for	ADP
ejde-942	110	23	the	the	DET
ejde-942	110	24	eigenpairs	eigenpair	NOUN
ejde-942	110	25	of	of	ADP
ejde-942	110	26	sturm	sturm	PROPN
ejde-942	110	27	-	-	PUNCT
ejde-942	110	28	liouville	liouville	NOUN
ejde-942	110	29	problems	problem	NOUN
ejde-942	110	30	in	in	ADP
ejde-942	110	31	inhomogeneous	inhomogeneous	ADJ
ejde-942	110	32	media	medium	NOUN
ejde-942	110	33	is	be	AUX
ejde-942	110	34	highly	highly	ADV
ejde-942	110	35	challenging	challenging	ADJ
ejde-942	110	36	,	,	PUNCT
ejde-942	110	37	so	so	CCONJ
ejde-942	110	38	the	the	DET
ejde-942	110	39	deep	deep	ADJ
ejde-942	110	40	learning	learning	NOUN
ejde-942	110	41	solution	solution	NOUN
ejde-942	110	42	we	we	PRON
ejde-942	110	43	designed	design	VERB
ejde-942	110	44	can	can	AUX
ejde-942	110	45	only	only	ADV
ejde-942	110	46	be	be	AUX
ejde-942	110	47	compared	compare	VERB
ejde-942	110	48	with	with	ADP
ejde-942	110	49	the	the	DET
ejde-942	110	50	classical	classical	ADJ
ejde-942	110	51	numerical	numerical	ADJ
ejde-942	110	52	method	method	NOUN
ejde-942	110	53	.	.	PUNCT
ejde-942	111	1	using	use	VERB
ejde-942	111	2	deep	deep	ADJ
ejde-942	111	3	learning	learning	NOUN
ejde-942	111	4	methods	method	NOUN
ejde-942	111	5	to	to	PART
ejde-942	111	6	solve	solve	VERB
ejde-942	111	7	eigenvalue	eigenvalue	NOUN
ejde-942	111	8	problems	problem	NOUN
ejde-942	111	9	offers	offer	VERB
ejde-942	111	10	numerous	numerous	ADJ
ejde-942	111	11	advantages	advantage	NOUN
ejde-942	111	12	.	.	PUNCT
ejde-942	112	1	one	one	NUM
ejde-942	112	2	such	such	ADJ
ejde-942	112	3	advantage	advantage	NOUN
ejde-942	112	4	is	be	AUX
ejde-942	112	5	that	that	SCONJ
ejde-942	112	6	the	the	DET
ejde-942	112	7	resulting	result	VERB
ejde-942	112	8	neural	neural	ADJ
ejde-942	112	9	network	network	NOUN
ejde-942	112	10	solutions	solution	NOUN
ejde-942	112	11	are	be	AUX
ejde-942	112	12	semi	semi	ADJ
ejde-942	112	13	-	-	ADJ
ejde-942	112	14	analytical	analytical	ADJ
ejde-942	112	15	,	,	PUNCT
ejde-942	112	16	meaning	mean	VERB
ejde-942	112	17	they	they	PRON
ejde-942	112	18	are	be	AUX
ejde-942	112	19	differentiable	differentiable	ADJ
ejde-942	112	20	.	.	PUNCT
ejde-942	113	1	the	the	DET
ejde-942	113	2	method	method	NOUN
ejde-942	113	3	is	be	AUX
ejde-942	113	4	also	also	ADV
ejde-942	113	5	simple	simple	ADJ
ejde-942	113	6	,	,	PUNCT
ejde-942	113	7	practical	practical	ADJ
ejde-942	113	8	,	,	PUNCT
ejde-942	113	9	and	and	CCONJ
ejde-942	113	10	easily	easily	ADV
ejde-942	113	11	ejde-2024/53	ejde-2024/53	ADP
ejde-942	113	12	deep	deep	ADJ
ejde-942	113	13	learning	learning	NOUN
ejde-942	113	14	method	method	NOUN
ejde-942	113	15	for	for	ADP
ejde-942	113	16	finding	find	VERB
ejde-942	113	17	eigenpairs	eigenpair	NOUN
ejde-942	113	18	5	5	NUM
ejde-942	113	19	extensible	extensible	ADJ
ejde-942	113	20	,	,	PUNCT
ejde-942	113	21	making	make	VERB
ejde-942	113	22	it	it	PRON
ejde-942	113	23	highly	highly	ADV
ejde-942	113	24	accessible	accessible	ADJ
ejde-942	113	25	to	to	ADP
ejde-942	113	26	non	non	NOUN
ejde-942	113	27	-	-	NOUN
ejde-942	113	28	experts	expert	NOUN
ejde-942	113	29	.	.	PUNCT
ejde-942	114	1	personally	personally	ADV
ejde-942	114	2	,	,	PUNCT
ejde-942	114	3	one	one	NUM
ejde-942	114	4	of	of	ADP
ejde-942	114	5	the	the	DET
ejde-942	114	6	greatest	great	ADJ
ejde-942	114	7	benefits	benefit	NOUN
ejde-942	114	8	i	i	PRON
ejde-942	114	9	have	have	AUX
ejde-942	114	10	found	find	VERB
ejde-942	114	11	is	be	AUX
ejde-942	114	12	that	that	SCONJ
ejde-942	114	13	for	for	ADP
ejde-942	114	14	derivative	derivative	ADJ
ejde-942	114	15	boundary	boundary	ADJ
ejde-942	114	16	value	value	NOUN
ejde-942	114	17	cases	case	NOUN
ejde-942	114	18	,	,	PUNCT
ejde-942	114	19	the	the	DET
ejde-942	114	20	neural	neural	ADJ
ejde-942	114	21	network	network	NOUN
ejde-942	114	22	solutions	solution	NOUN
ejde-942	114	23	obtained	obtain	VERB
ejde-942	114	24	through	through	ADP
ejde-942	114	25	this	this	DET
ejde-942	114	26	dimension	dimension	NOUN
ejde-942	114	27	-	-	PUNCT
ejde-942	114	28	free	free	ADJ
ejde-942	114	29	approach	approach	NOUN
ejde-942	114	30	are	be	AUX
ejde-942	114	31	self	self	NOUN
ejde-942	114	32	-	-	PUNCT
ejde-942	114	33	adjoint	adjoint	NOUN
ejde-942	114	34	,	,	PUNCT
ejde-942	114	35	aligning	align	VERB
ejde-942	114	36	with	with	ADP
ejde-942	114	37	the	the	DET
ejde-942	114	38	original	original	ADJ
ejde-942	114	39	problem	problem	NOUN
ejde-942	114	40	.	.	PUNCT
ejde-942	115	1	in	in	ADP
ejde-942	115	2	contrast	contrast	NOUN
ejde-942	115	3	,	,	PUNCT
ejde-942	115	4	classical	classical	ADJ
ejde-942	115	5	numerical	numerical	ADJ
ejde-942	115	6	methods	method	NOUN
ejde-942	115	7	,	,	PUNCT
ejde-942	115	8	such	such	ADJ
ejde-942	115	9	as	as	ADP
ejde-942	115	10	finite	finite	ADJ
ejde-942	115	11	difference	difference	NOUN
ejde-942	115	12	method	method	NOUN
ejde-942	115	13	,	,	PUNCT
ejde-942	115	14	often	often	ADV
ejde-942	115	15	yield	yield	VERB
ejde-942	115	16	approximate	approximate	ADJ
ejde-942	115	17	solutions	solution	NOUN
ejde-942	115	18	that	that	PRON
ejde-942	115	19	do	do	AUX
ejde-942	115	20	not	not	PART
ejde-942	115	21	satisfy	satisfy	VERB
ejde-942	115	22	self	self	NOUN
ejde-942	115	23	-	-	PUNCT
ejde-942	115	24	adjointness	adjointness	NOUN
ejde-942	115	25	,	,	PUNCT
ejde-942	115	26	leading	lead	VERB
ejde-942	115	27	to	to	ADP
ejde-942	115	28	numerical	numerical	ADJ
ejde-942	115	29	eigenfunctions	eigenfunction	NOUN
ejde-942	115	30	that	that	PRON
ejde-942	115	31	are	be	AUX
ejde-942	115	32	not	not	PART
ejde-942	115	33	orthogonal	orthogonal	ADJ
ejde-942	115	34	,	,	PUNCT
ejde-942	115	35	see	see	VERB
ejde-942	115	36	[	[	X
ejde-942	115	37	1	1	NUM
ejde-942	115	38	]	]	PUNCT
ejde-942	115	39	.	.	PUNCT
ejde-942	116	1	using	use	VERB
ejde-942	116	2	neural	neural	ADJ
ejde-942	116	3	networks	network	NOUN
ejde-942	116	4	to	to	PART
ejde-942	116	5	find	find	VERB
ejde-942	116	6	the	the	DET
ejde-942	116	7	eigenpairs	eigenpair	NOUN
ejde-942	116	8	of	of	ADP
ejde-942	116	9	{	{	PUNCT
ejde-942	116	10	λk	λk	INTJ
ejde-942	116	11	,	,	PUNCT
ejde-942	116	12	φk	φk	ADP
ejde-942	116	13	}	}	PUNCT
ejde-942	116	14	in	in	ADP
ejde-942	116	15	inhomogeneous	inhomogeneous	ADJ
ejde-942	116	16	media	medium	NOUN
ejde-942	116	17	only	only	ADV
ejde-942	116	18	requires	require	VERB
ejde-942	116	19	starting	start	VERB
ejde-942	116	20	with	with	ADP
ejde-942	116	21	the	the	DET
ejde-942	116	22	problem	problem	NOUN
ejde-942	116	23	itself	itself	PRON
ejde-942	116	24	.	.	PUNCT
ejde-942	117	1	we	we	PRON
ejde-942	117	2	gradually	gradually	ADV
ejde-942	117	3	learn	learn	VERB
ejde-942	117	4	eigenpairs	eigenpair	NOUN
ejde-942	117	5	,	,	PUNCT
ejde-942	117	6	starting	start	VERB
ejde-942	117	7	from	from	ADP
ejde-942	117	8	smaller	small	ADJ
ejde-942	117	9	values	value	NOUN
ejde-942	117	10	of	of	ADP
ejde-942	117	11	k	k	PROPN
ejde-942	117	12	and	and	CCONJ
ejde-942	117	13	progressing	progress	VERB
ejde-942	117	14	to	to	ADP
ejde-942	117	15	larger	large	ADJ
ejde-942	117	16	ones	one	NOUN
ejde-942	117	17	.	.	PUNCT
ejde-942	118	1	inspired	inspire	VERB
ejde-942	118	2	by	by	ADP
ejde-942	118	3	the	the	DET
ejde-942	118	4	pinn	pinn	NOUN
ejde-942	118	5	,	,	PUNCT
ejde-942	118	6	we	we	PRON
ejde-942	118	7	integrate	integrate	VERB
ejde-942	118	8	the	the	DET
ejde-942	118	9	equation	equation	NOUN
ejde-942	118	10	equalities	equality	NOUN
ejde-942	118	11	,	,	PUNCT
ejde-942	118	12	boundary	boundary	ADJ
ejde-942	118	13	value	value	NOUN
ejde-942	118	14	conditions	condition	NOUN
ejde-942	118	15	,	,	PUNCT
ejde-942	118	16	normalization	normalization	NOUN
ejde-942	118	17	of	of	ADP
ejde-942	118	18	eigenfunctions	eigenfunction	NOUN
ejde-942	118	19	,	,	PUNCT
ejde-942	118	20	and	and	CCONJ
ejde-942	118	21	orthogonality	orthogonality	NOUN
ejde-942	118	22	among	among	ADP
ejde-942	118	23	distinct	distinct	ADJ
ejde-942	118	24	eigenfunctions	eigenfunction	NOUN
ejde-942	118	25	into	into	ADP
ejde-942	118	26	the	the	DET
ejde-942	118	27	loss	loss	NOUN
ejde-942	118	28	function	function	NOUN
ejde-942	118	29	.	.	PUNCT
ejde-942	119	1	this	this	DET
ejde-942	119	2	approach	approach	NOUN
ejde-942	119	3	ensures	ensure	VERB
ejde-942	119	4	the	the	DET
ejde-942	119	5	feasibility	feasibility	NOUN
ejde-942	119	6	and	and	CCONJ
ejde-942	119	7	practical	practical	ADJ
ejde-942	119	8	implementation	implementation	NOUN
ejde-942	119	9	of	of	ADP
ejde-942	119	10	our	our	PRON
ejde-942	119	11	method	method	NOUN
ejde-942	119	12	.	.	PUNCT
ejde-942	120	1	specifically	specifically	ADV
ejde-942	120	2	,	,	PUNCT
ejde-942	120	3	we	we	PRON
ejde-942	120	4	introduce	introduce	VERB
ejde-942	120	5	a	a	DET
ejde-942	120	6	parameterized	parameterized	ADJ
ejde-942	120	7	form	form	NOUN
ejde-942	120	8	of	of	ADP
ejde-942	120	9	the	the	DET
ejde-942	120	10	k	k	NOUN
ejde-942	120	11	-	-	PUNCT
ejde-942	120	12	th	th	VERB
ejde-942	120	13	eigenfunction	eigenfunction	NOUN
ejde-942	120	14	,	,	PUNCT
ejde-942	120	15	denoted	denote	VERB
ejde-942	120	16	by	by	ADP
ejde-942	120	17	φθk	φθk	NOUN
ejde-942	120	18	.	.	PUNCT
ejde-942	121	1	in	in	ADP
ejde-942	121	2	this	this	DET
ejde-942	121	3	context	context	NOUN
ejde-942	121	4	,	,	PUNCT
ejde-942	121	5	θk	θk	NOUN
ejde-942	121	6	functions	function	NOUN
ejde-942	121	7	as	as	ADP
ejde-942	121	8	the	the	DET
ejde-942	121	9	primary	primary	ADJ
ejde-942	121	10	training	training	NOUN
ejde-942	121	11	parameter	parameter	NOUN
ejde-942	121	12	.	.	PUNCT
ejde-942	122	1	moreover	moreover	ADV
ejde-942	122	2	,	,	PUNCT
ejde-942	122	3	within	within	ADP
ejde-942	122	4	our	our	PRON
ejde-942	122	5	comprehensive	comprehensive	ADJ
ejde-942	122	6	framework	framework	NOUN
ejde-942	122	7	,	,	PUNCT
ejde-942	122	8	we	we	PRON
ejde-942	122	9	incorporate	incorporate	VERB
ejde-942	122	10	the	the	DET
ejde-942	122	11	eigenvalue	eigenvalue	NOUN
ejde-942	122	12	λk	λk	ADP
ejde-942	122	13	as	as	ADP
ejde-942	122	14	an	an	DET
ejde-942	122	15	additional	additional	ADJ
ejde-942	122	16	parameter	parameter	NOUN
ejde-942	122	17	that	that	PRON
ejde-942	122	18	undergoes	undergo	VERB
ejde-942	122	19	training	training	NOUN
ejde-942	122	20	alongside	alongside	ADP
ejde-942	122	21	θk	θk	NOUN
ejde-942	122	22	.	.	PUNCT
ejde-942	123	1	for	for	ADP
ejde-942	123	2	each	each	DET
ejde-942	123	3	k	k	NOUN
ejde-942	123	4	,	,	PUNCT
ejde-942	123	5	when	when	SCONJ
ejde-942	123	6	φθk	φθk	NOUN
ejde-942	123	7	learns	learn	VERB
ejde-942	123	8	the	the	DET
ejde-942	123	9	expected	expect	VERB
ejde-942	123	10	target	target	NOUN
ejde-942	123	11	through	through	ADP
ejde-942	123	12	training	training	NOUN
ejde-942	123	13	,	,	PUNCT
ejde-942	123	14	λk	λk	PROPN
ejde-942	123	15	will	will	AUX
ejde-942	123	16	also	also	ADV
ejde-942	123	17	be	be	AUX
ejde-942	123	18	automatically	automatically	ADV
ejde-942	123	19	updated	update	VERB
ejde-942	123	20	with	with	ADP
ejde-942	123	21	the	the	DET
ejde-942	123	22	network	network	NOUN
ejde-942	123	23	to	to	ADP
ejde-942	123	24	the	the	DET
ejde-942	123	25	eigenvalue	eigenvalue	NOUN
ejde-942	123	26	corresponding	corresponding	NOUN
ejde-942	123	27	to	to	ADP
ejde-942	123	28	the	the	DET
ejde-942	123	29	eigenfunction	eigenfunction	NOUN
ejde-942	123	30	network	network	NOUN
ejde-942	123	31	φθk	φθk	NOUN
ejde-942	123	32	.	.	PUNCT
ejde-942	124	1	this	this	DET
ejde-942	124	2	design	design	NOUN
ejde-942	124	3	greatly	greatly	ADV
ejde-942	124	4	reduces	reduce	VERB
ejde-942	124	5	the	the	DET
ejde-942	124	6	complexity	complexity	NOUN
ejde-942	124	7	of	of	ADP
ejde-942	124	8	the	the	DET
ejde-942	124	9	network	network	NOUN
ejde-942	124	10	and	and	CCONJ
ejde-942	124	11	the	the	DET
ejde-942	124	12	difficulty	difficulty	NOUN
ejde-942	124	13	of	of	ADP
ejde-942	124	14	calculation	calculation	NOUN
ejde-942	124	15	,	,	PUNCT
ejde-942	124	16	so	so	SCONJ
ejde-942	124	17	that	that	SCONJ
ejde-942	124	18	we	we	PRON
ejde-942	124	19	only	only	ADV
ejde-942	124	20	need	need	VERB
ejde-942	124	21	to	to	PART
ejde-942	124	22	pay	pay	VERB
ejde-942	124	23	attention	attention	NOUN
ejde-942	124	24	to	to	ADP
ejde-942	124	25	the	the	DET
ejde-942	124	26	training	training	NOUN
ejde-942	124	27	of	of	ADP
ejde-942	124	28	the	the	DET
ejde-942	124	29	eigenfunction	eigenfunction	NOUN
ejde-942	124	30	network	network	NOUN
ejde-942	124	31	,	,	PUNCT
ejde-942	124	32	and	and	CCONJ
ejde-942	124	33	do	do	AUX
ejde-942	124	34	not	not	PART
ejde-942	124	35	need	need	VERB
ejde-942	124	36	to	to	PART
ejde-942	124	37	design	design	VERB
ejde-942	124	38	an	an	DET
ejde-942	124	39	additional	additional	ADJ
ejde-942	124	40	network	network	NOUN
ejde-942	124	41	to	to	PART
ejde-942	124	42	train	train	VERB
ejde-942	124	43	the	the	DET
ejde-942	124	44	eigenvalue	eigenvalue	NOUN
ejde-942	124	45	.	.	PUNCT
ejde-942	124	46	to	to	PART
ejde-942	124	47	optimize	optimize	VERB
ejde-942	124	48	the	the	DET
ejde-942	124	49	neural	neural	ADJ
ejde-942	124	50	network	network	NOUN
ejde-942	124	51	φθk	φθk	NOUN
ejde-942	124	52	with	with	ADP
ejde-942	124	53	respect	respect	NOUN
ejde-942	124	54	to	to	ADP
ejde-942	124	55	its	its	PRON
ejde-942	124	56	parameter	parameter	NOUN
ejde-942	124	57	θk	θk	NOUN
ejde-942	124	58	,	,	PUNCT
ejde-942	124	59	we	we	PRON
ejde-942	124	60	define	define	VERB
ejde-942	124	61	the	the	DET
ejde-942	124	62	loss	loss	NOUN
ejde-942	124	63	function	function	NOUN
ejde-942	124	64	j(θk	j(θk	PROPN
ejde-942	124	65	,	,	PUNCT
ejde-942	124	66	λk	λk	X
ejde-942	124	67	)	)	PUNCT
ejde-942	124	68	=	=	SYM
ejde-942	124	69	αjeq(θk	αjeq(θk	PROPN
ejde-942	124	70	,	,	PUNCT
ejde-942	124	71	λk	λk	PROPN
ejde-942	124	72	)	)	PUNCT
ejde-942	125	1	+	+	CCONJ
ejde-942	125	2	βjbdry(θk	βjbdry(θk	NOUN
ejde-942	125	3	)	)	PUNCT
ejde-942	125	4	+	+	X
ejde-942	125	5	γjnor(θk	γjnor(θk	NOUN
ejde-942	125	6	)	)	PUNCT
ejde-942	125	7	+	+	SYM
ejde-942	125	8	δkjorth(θk	δkjorth(θk	NOUN
ejde-942	125	9	)	)	PUNCT
ejde-942	125	10	,	,	PUNCT
ejde-942	125	11	(	(	PUNCT
ejde-942	125	12	2.6	2.6	NUM
ejde-942	125	13	)	)	PUNCT
ejde-942	125	14	where	where	SCONJ
ejde-942	125	15	α	α	X
ejde-942	125	16	,	,	PUNCT
ejde-942	125	17	β	β	X
ejde-942	125	18	,	,	PUNCT
ejde-942	125	19	γ	γ	PROPN
ejde-942	125	20	,	,	PUNCT
ejde-942	125	21	δk	δk	X
ejde-942	125	22	are	be	AUX
ejde-942	125	23	hyper	hyper	ADJ
ejde-942	125	24	-	-	NOUN
ejde-942	125	25	parameters	parameter	NOUN
ejde-942	125	26	that	that	PRON
ejde-942	125	27	control	control	VERB
ejde-942	125	28	the	the	DET
ejde-942	125	29	relative	relative	ADJ
ejde-942	125	30	importance	importance	NOUN
ejde-942	125	31	of	of	ADP
ejde-942	125	32	each	each	DET
ejde-942	125	33	term	term	NOUN
ejde-942	125	34	.	.	PUNCT
ejde-942	126	1	the	the	DET
ejde-942	126	2	individual	individual	ADJ
ejde-942	126	3	terms	term	NOUN
ejde-942	126	4	are	be	AUX
ejde-942	126	5	as	as	SCONJ
ejde-942	126	6	follows	follow	VERB
ejde-942	126	7	:	:	PUNCT
ejde-942	126	8	jeq(θk	jeq(θk	NOUN
ejde-942	126	9	,	,	PUNCT
ejde-942	126	10	λk	λk	PROPN
ejde-942	126	11	)	)	PUNCT
ejde-942	126	12	=	=	PUNCT
ejde-942	126	13	∥l[φθk	∥l[φθk	PUNCT
ejde-942	126	14	]	]	PUNCT
ejde-942	126	15	−	−	X
ejde-942	126	16	λkφθk∥2σ	λkφθk∥2σ	X
ejde-942	126	17	(	(	PUNCT
ejde-942	126	18	2.7	2.7	NUM
ejde-942	126	19	)	)	PUNCT
ejde-942	126	20	enforces	enforce	VERB
ejde-942	126	21	the	the	DET
ejde-942	126	22	constraints	constraint	NOUN
ejde-942	126	23	imposed	impose	VERB
ejde-942	126	24	by	by	ADP
ejde-942	126	25	the	the	DET
ejde-942	126	26	equation	equation	NOUN
ejde-942	126	27	.	.	PUNCT
ejde-942	127	1	jbdry(θk	jbdry(θk	ADJ
ejde-942	127	2	)	)	PUNCT
ejde-942	127	3	=	=	SYM
ejde-942	128	1	∣∣b[φθk	∣∣b[φθk	PROPN
ejde-942	128	2	]	]	PUNCT
ejde-942	128	3	∣∣2	∣∣2	PROPN
ejde-942	128	4	(	(	PUNCT
ejde-942	128	5	2.8	2.8	NUM
ejde-942	128	6	)	)	PUNCT
ejde-942	128	7	accounts	account	NOUN
ejde-942	128	8	for	for	ADP
ejde-942	128	9	the	the	DET
ejde-942	128	10	constraints	constraint	NOUN
ejde-942	128	11	imposed	impose	VERB
ejde-942	128	12	by	by	ADP
ejde-942	128	13	the	the	DET
ejde-942	128	14	boundary	boundary	ADJ
ejde-942	128	15	value	value	NOUN
ejde-942	128	16	conditions	condition	NOUN
ejde-942	128	17	.	.	PUNCT
ejde-942	129	1	jnor(θk	jnor(θk	NOUN
ejde-942	129	2	)	)	PUNCT
ejde-942	129	3	=	=	PUNCT
ejde-942	130	1	(	(	PUNCT
ejde-942	130	2	∥φθk∥2σ	∥φθk∥2σ	VERB
ejde-942	130	3	−	−	PROPN
ejde-942	130	4	1	1	NUM
ejde-942	130	5	)	)	SYM
ejde-942	130	6	2	2	NUM
ejde-942	130	7	∥φθk∥2σ	∥φθk∥2σ	PROPN
ejde-942	130	8	,	,	PUNCT
ejde-942	130	9	(	(	PUNCT
ejde-942	130	10	2.9	2.9	NUM
ejde-942	130	11	)	)	PUNCT
ejde-942	130	12	is	be	AUX
ejde-942	130	13	responsible	responsible	ADJ
ejde-942	130	14	for	for	ADP
ejde-942	130	15	the	the	DET
ejde-942	130	16	normalization	normalization	NOUN
ejde-942	130	17	of	of	ADP
ejde-942	130	18	the	the	DET
ejde-942	130	19	eigenfunction	eigenfunction	NOUN
ejde-942	130	20	in	in	ADP
ejde-942	130	21	the	the	DET
ejde-942	130	22	weight	weight	NOUN
ejde-942	130	23	space	space	NOUN
ejde-942	130	24	.	.	PUNCT
ejde-942	131	1	the	the	DET
ejde-942	131	2	denominator	denominator	NOUN
ejde-942	131	3	is	be	AUX
ejde-942	131	4	included	include	VERB
ejde-942	131	5	to	to	PART
ejde-942	131	6	prevent	prevent	VERB
ejde-942	131	7	the	the	DET
ejde-942	131	8	norm	norm	NOUN
ejde-942	131	9	of	of	ADP
ejde-942	131	10	the	the	DET
ejde-942	131	11	eigenfunction	eigenfunction	NOUN
ejde-942	131	12	from	from	ADP
ejde-942	131	13	vanishing	vanish	VERB
ejde-942	131	14	during	during	ADP
ejde-942	131	15	the	the	DET
ejde-942	131	16	computation	computation	NOUN
ejde-942	131	17	of	of	ADP
ejde-942	131	18	higher	high	ADJ
ejde-942	131	19	eigenvalues	eigenvalue	NOUN
ejde-942	131	20	.	.	PUNCT
ejde-942	132	1	jorth(θk	jorth(θk	ADJ
ejde-942	132	2	)	)	PUNCT
ejde-942	132	3	=	=	PUNCT
ejde-942	133	1	k−1∑	k−1∑	PROPN
ejde-942	133	2	j=1	j=1	NOUN
ejde-942	133	3	∣∣(φθk	∣∣(φθk	NOUN
ejde-942	133	4	,	,	PUNCT
ejde-942	133	5	φθj	φθj	NOUN
ejde-942	133	6	)	)	PUNCT
ejde-942	133	7	σ	σ	PROPN
ejde-942	133	8	∣∣2	∣∣2	PROPN
ejde-942	133	9	,	,	PUNCT
ejde-942	133	10	(	(	PUNCT
ejde-942	133	11	2.10	2.10	NUM
ejde-942	133	12	)	)	PUNCT
ejde-942	133	13	ensures	ensure	VERB
ejde-942	133	14	orthogonality	orthogonality	NOUN
ejde-942	133	15	of	of	ADP
ejde-942	133	16	the	the	DET
ejde-942	133	17	current	current	ADJ
ejde-942	133	18	eigenfunction	eigenfunction	NOUN
ejde-942	133	19	with	with	ADP
ejde-942	133	20	respect	respect	NOUN
ejde-942	133	21	to	to	ADP
ejde-942	133	22	previously	previously	ADV
ejde-942	133	23	computed	compute	VERB
ejde-942	133	24	eigenfunctions	eigenfunction	NOUN
ejde-942	133	25	in	in	ADP
ejde-942	133	26	the	the	DET
ejde-942	133	27	weight	weight	NOUN
ejde-942	133	28	space	space	NOUN
ejde-942	133	29	.	.	PUNCT
ejde-942	134	1	under	under	ADP
ejde-942	134	2	this	this	DET
ejde-942	134	3	framework	framework	NOUN
ejde-942	134	4	,	,	PUNCT
ejde-942	134	5	the	the	DET
ejde-942	134	6	choice	choice	NOUN
ejde-942	134	7	of	of	ADP
ejde-942	134	8	δk	δk	PRON
ejde-942	134	9	is	be	AUX
ejde-942	134	10	crucial	crucial	ADJ
ejde-942	134	11	for	for	SCONJ
ejde-942	134	12	accurately	accurately	ADV
ejde-942	134	13	computing	compute	VERB
ejde-942	134	14	higherorder	higherorder	NOUN
ejde-942	134	15	eigenvalues	eigenvalue	VERB
ejde-942	134	16	λk	λk	ADP
ejde-942	134	17	.	.	PUNCT
ejde-942	135	1	when	when	SCONJ
ejde-942	135	2	the	the	DET
ejde-942	135	3	epoch	epoch	NOUN
ejde-942	135	4	is	be	AUX
ejde-942	135	5	less	less	ADJ
ejde-942	135	6	than	than	ADP
ejde-942	135	7	half	half	NOUN
ejde-942	135	8	of	of	ADP
ejde-942	135	9	the	the	DET
ejde-942	135	10	total	total	ADJ
ejde-942	135	11	number	number	NOUN
ejde-942	135	12	of	of	ADP
ejde-942	135	13	epochs	epoch	NOUN
ejde-942	135	14	,	,	PUNCT
ejde-942	135	15	we	we	PRON
ejde-942	135	16	set	set	VERB
ejde-942	135	17	δk	δk	ADP
ejde-942	135	18	equal	equal	ADJ
ejde-942	135	19	to	to	ADP
ejde-942	135	20	10	10	NUM
ejde-942	135	21	·	·	PUNCT
ejde-942	135	22	λk−1	λk−1	NOUN
ejde-942	135	23	to	to	PART
ejde-942	135	24	ensure	ensure	VERB
ejde-942	135	25	that	that	SCONJ
ejde-942	135	26	the	the	DET
ejde-942	135	27	eigenvalues	eigenvalue	NOUN
ejde-942	135	28	can	can	AUX
ejde-942	135	29	correctly	correctly	ADV
ejde-942	135	30	6	6	NUM
ejde-942	135	31	s.	s.	PROPN
ejde-942	135	32	zhang	zhang	PROPN
ejde-942	135	33	,	,	PUNCT
ejde-942	135	34	j.	j.	PROPN
ejde-942	135	35	zu	zu	PROPN
ejde-942	135	36	,	,	PUNCT
ejde-942	135	37	j.	j.	PROPN
ejde-942	135	38	zhang	zhang	PROPN
ejde-942	135	39	ejde-2024/53	ejde-2024/53	PROPN
ejde-942	135	40	approach	approach	VERB
ejde-942	135	41	the	the	DET
ejde-942	135	42	desired	desire	VERB
ejde-942	135	43	values	value	NOUN
ejde-942	135	44	.	.	PUNCT
ejde-942	136	1	however	however	ADV
ejde-942	136	2	,	,	PUNCT
ejde-942	136	3	when	when	SCONJ
ejde-942	136	4	the	the	DET
ejde-942	136	5	epoch	epoch	NOUN
ejde-942	136	6	exceeds	exceed	VERB
ejde-942	136	7	half	half	NOUN
ejde-942	136	8	of	of	ADP
ejde-942	136	9	the	the	DET
ejde-942	136	10	total	total	NOUN
ejde-942	136	11	,	,	PUNCT
ejde-942	136	12	we	we	PRON
ejde-942	136	13	set	set	VERB
ejde-942	136	14	δk	δk	PUNCT
ejde-942	136	15	to	to	ADP
ejde-942	136	16	1	1	NUM
ejde-942	136	17	to	to	PART
ejde-942	136	18	reduce	reduce	VERB
ejde-942	136	19	the	the	DET
ejde-942	136	20	weight	weight	NOUN
ejde-942	136	21	of	of	ADP
ejde-942	136	22	the	the	DET
ejde-942	136	23	orthogonality	orthogonality	NOUN
ejde-942	136	24	term	term	NOUN
ejde-942	136	25	,	,	PUNCT
ejde-942	136	26	thereby	thereby	ADV
ejde-942	136	27	facilitating	facilitate	VERB
ejde-942	136	28	better	well	ADJ
ejde-942	136	29	computation	computation	NOUN
ejde-942	136	30	of	of	ADP
ejde-942	136	31	the	the	DET
ejde-942	136	32	loss	loss	NOUN
ejde-942	136	33	associated	associate	VERB
ejde-942	136	34	with	with	ADP
ejde-942	136	35	the	the	DET
ejde-942	136	36	equation	equation	NOUN
ejde-942	136	37	term	term	NOUN
ejde-942	136	38	.	.	PUNCT
ejde-942	137	1	note	note	VERB
ejde-942	137	2	that	that	SCONJ
ejde-942	137	3	both	both	CCONJ
ejde-942	137	4	the	the	DET
ejde-942	137	5	divisor	divisor	NOUN
ejde-942	137	6	of	of	ADP
ejde-942	137	7	the	the	DET
ejde-942	137	8	third	third	ADJ
ejde-942	137	9	term	term	NOUN
ejde-942	137	10	and	and	CCONJ
ejde-942	137	11	the	the	DET
ejde-942	137	12	addition	addition	NOUN
ejde-942	137	13	of	of	ADP
ejde-942	137	14	the	the	DET
ejde-942	137	15	δk	δk	NOUN
ejde-942	137	16	are	be	AUX
ejde-942	137	17	specially	specially	ADV
ejde-942	137	18	designed	design	VERB
ejde-942	137	19	for	for	ADP
ejde-942	137	20	obtaining	obtain	VERB
ejde-942	137	21	the	the	DET
ejde-942	137	22	higher	high	ADJ
ejde-942	137	23	eigenpairs	eigenpair	NOUN
ejde-942	137	24	.	.	PUNCT
ejde-942	138	1	for	for	ADP
ejde-942	138	2	details	detail	NOUN
ejde-942	138	3	of	of	ADP
ejde-942	138	4	the	the	DET
ejde-942	138	5	process	process	NOUN
ejde-942	138	6	,	,	PUNCT
ejde-942	138	7	see	see	VERB
ejde-942	138	8	algorithm	algorithm	NOUN
ejde-942	138	9	1	1	NUM
ejde-942	138	10	.	.	PUNCT
ejde-942	139	1	algorithm	algorithm	NOUN
ejde-942	139	2	1	1	NUM
ejde-942	139	3	deep	deep	ADJ
ejde-942	139	4	learning	learning	NOUN
ejde-942	139	5	method	method	NOUN
ejde-942	139	6	for	for	ADP
ejde-942	139	7	finding	find	VERB
ejde-942	139	8	m	m	NOUN
ejde-942	139	9	eigenpairs	eigenpair	NOUN
ejde-942	139	10	1	1	NUM
ejde-942	139	11	:	:	PUNCT
ejde-942	139	12	for	for	ADP
ejde-942	139	13	k	k	PROPN
ejde-942	139	14	=	=	SYM
ejde-942	139	15	1	1	NUM
ejde-942	139	16	to	to	PART
ejde-942	139	17	m	m	VERB
ejde-942	139	18	do	do	VERB
ejde-942	139	19	2	2	NUM
ejde-942	139	20	:	:	PUNCT
ejde-942	139	21	set	set	VERB
ejde-942	139	22	total	total	ADJ
ejde-942	139	23	number	number	NOUN
ejde-942	139	24	of	of	ADP
ejde-942	139	25	epochs	epoch	NOUN
ejde-942	139	26	nepochs	nepoch	VERB
ejde-942	139	27	3	3	NUM
ejde-942	139	28	:	:	PUNCT
ejde-942	139	29	define	define	VERB
ejde-942	139	30	hyper	hyper	NOUN
ejde-942	139	31	-	-	NOUN
ejde-942	139	32	parameters	parameter	NOUN
ejde-942	139	33	α	α	NOUN
ejde-942	139	34	,	,	PUNCT
ejde-942	139	35	β	β	X
ejde-942	139	36	,	,	PUNCT
ejde-942	139	37	γ	γ	X
ejde-942	139	38	,	,	PUNCT
ejde-942	139	39	λ0	λ0	NOUN
ejde-942	139	40	4	4	NUM
ejde-942	139	41	:	:	PUNCT
ejde-942	139	42	initialize	initialize	VERB
ejde-942	139	43	neural	neural	ADJ
ejde-942	139	44	network	network	NOUN
ejde-942	139	45	φθk	φθk	NOUN
ejde-942	139	46	with	with	ADP
ejde-942	139	47	parameters	parameter	NOUN
ejde-942	139	48	θk	θk	VERB
ejde-942	139	49	and	and	CCONJ
ejde-942	139	50	λk	λk	NOUN
ejde-942	140	1	=	=	PUNCT
ejde-942	140	2	λk−1	λk−1	NOUN
ejde-942	140	3	5	5	NUM
ejde-942	140	4	:	:	PUNCT
ejde-942	140	5	for	for	ADP
ejde-942	140	6	epoch	epoch	NOUN
ejde-942	140	7	=	=	SYM
ejde-942	140	8	1	1	NUM
ejde-942	140	9	to	to	ADP
ejde-942	140	10	nepochs	nepoch	NOUN
ejde-942	140	11	do	do	VERB
ejde-942	140	12	6	6	NUM
ejde-942	140	13	:	:	PUNCT
ejde-942	140	14	if	if	SCONJ
ejde-942	140	15	epoch	epoch	PROPN
ejde-942	140	16	<	<	X
ejde-942	140	17	nepochs	nepochs	PROPN
ejde-942	140	18	2	2	NUM
ejde-942	140	19	then	then	ADV
ejde-942	140	20	7	7	NUM
ejde-942	140	21	:	:	PUNCT
ejde-942	140	22	set	set	VERB
ejde-942	140	23	δk	δk	PRON
ejde-942	141	1	=	=	SYM
ejde-942	141	2	10	10	NUM
ejde-942	141	3	·	·	PUNCT
ejde-942	141	4	λk−1	λk−1	NOUN
ejde-942	141	5	8	8	NUM
ejde-942	141	6	:	:	PUNCT
ejde-942	141	7	else	else	ADV
ejde-942	141	8	9	9	NUM
ejde-942	141	9	:	:	PUNCT
ejde-942	141	10	set	set	VERB
ejde-942	141	11	δk	δk	PRON
ejde-942	141	12	=	=	NOUN
ejde-942	141	13	1	1	NUM
ejde-942	141	14	10	10	NUM
ejde-942	141	15	:	:	PUNCT
ejde-942	141	16	end	end	VERB
ejde-942	141	17	if	if	SCONJ
ejde-942	141	18	11	11	NUM
ejde-942	141	19	:	:	PUNCT
ejde-942	141	20	compute	compute	NOUN
ejde-942	141	21	loss	loss	NOUN
ejde-942	141	22	function	function	NOUN
ejde-942	141	23	j(θk	j(θk	PROPN
ejde-942	141	24	,	,	PUNCT
ejde-942	141	25	λk	λk	ADV
ejde-942	141	26	)	)	PUNCT
ejde-942	141	27	using	use	VERB
ejde-942	141	28	eq	eq	X
ejde-942	141	29	.	.	PUNCT
ejde-942	142	1	(	(	PUNCT
ejde-942	142	2	2.6	2.6	NUM
ejde-942	142	3	)	)	PUNCT
ejde-942	142	4	12	12	NUM
ejde-942	142	5	:	:	PUNCT
ejde-942	142	6	compute	compute	NOUN
ejde-942	142	7	gradients	gradient	NOUN
ejde-942	142	8	of	of	ADP
ejde-942	142	9	j(θk	j(θk	PROPN
ejde-942	142	10	,	,	PUNCT
ejde-942	142	11	λk	λk	X
ejde-942	142	12	)	)	PUNCT
ejde-942	142	13	with	with	ADP
ejde-942	142	14	respect	respect	NOUN
ejde-942	142	15	to	to	ADP
ejde-942	142	16	θk	θk	NOUN
ejde-942	142	17	and	and	CCONJ
ejde-942	142	18	λk	λk	ADP
ejde-942	142	19	13	13	NUM
ejde-942	142	20	:	:	PUNCT
ejde-942	142	21	update	update	NOUN
ejde-942	142	22	θk	θk	NOUN
ejde-942	142	23	and	and	CCONJ
ejde-942	142	24	λk	λk	X
ejde-942	142	25	using	use	VERB
ejde-942	142	26	gradient	gradient	ADJ
ejde-942	142	27	descent	descent	NOUN
ejde-942	142	28	optimizer	optimizer	NOUN
ejde-942	142	29	14	14	NUM
ejde-942	142	30	:	:	PUNCT
ejde-942	142	31	if	if	SCONJ
ejde-942	142	32	convergence	convergence	NOUN
ejde-942	142	33	criterion	criterion	NOUN
ejde-942	142	34	is	be	AUX
ejde-942	142	35	met	meet	VERB
ejde-942	142	36	then	then	ADV
ejde-942	142	37	15	15	NUM
ejde-942	142	38	:	:	PUNCT
ejde-942	142	39	break	break	VERB
ejde-942	142	40	the	the	DET
ejde-942	142	41	loop	loop	NOUN
ejde-942	142	42	16	16	NUM
ejde-942	142	43	:	:	PUNCT
ejde-942	142	44	end	end	VERB
ejde-942	142	45	if	if	SCONJ
ejde-942	142	46	17	17	NUM
ejde-942	142	47	:	:	PUNCT
ejde-942	142	48	end	end	VERB
ejde-942	142	49	for	for	ADP
ejde-942	142	50	18	18	NUM
ejde-942	142	51	:	:	PUNCT
ejde-942	142	52	output	output	NOUN
ejde-942	142	53	optimized	optimize	VERB
ejde-942	142	54	parameters	parameter	NOUN
ejde-942	142	55	θk	θk	VERB
ejde-942	142	56	and	and	CCONJ
ejde-942	142	57	eigenvalue	eigenvalue	PROPN
ejde-942	142	58	λk	λk	ADP
ejde-942	142	59	19	19	NUM
ejde-942	142	60	:	:	PUNCT
ejde-942	142	61	end	end	VERB
ejde-942	142	62	for	for	ADP
ejde-942	142	63	in	in	ADP
ejde-942	142	64	the	the	DET
ejde-942	142	65	individual	individual	ADJ
ejde-942	142	66	terms	term	NOUN
ejde-942	142	67	of	of	ADP
ejde-942	142	68	cost	cost	NOUN
ejde-942	142	69	function	function	NOUN
ejde-942	142	70	,	,	PUNCT
ejde-942	142	71	jeq(θk	jeq(θk	PROPN
ejde-942	142	72	,	,	PUNCT
ejde-942	142	73	λk	λk	X
ejde-942	142	74	)	)	PUNCT
ejde-942	142	75	is	be	AUX
ejde-942	142	76	calculated	calculate	VERB
ejde-942	142	77	by	by	ADP
ejde-942	142	78	∥l[φθk	∥l[φθk	PRON
ejde-942	142	79	]	]	PUNCT
ejde-942	142	80	−	−	X
ejde-942	142	81	λkφθk∥2σ	λkφθk∥2σ	NOUN
ejde-942	142	82	=	=	SYM
ejde-942	142	83	1	1	NUM
ejde-942	142	84	nk	nk	NOUN
ejde-942	142	85	nk∑	nk∑	PROPN
ejde-942	142	86	j=1	j=1	PROPN
ejde-942	142	87	σ	σ	PROPN
ejde-942	142	88	(	(	PUNCT
ejde-942	142	89	x(j)r	x(j)r	PROPN
ejde-942	142	90	)	)	PUNCT
ejde-942	142	91	(	(	PUNCT
ejde-942	142	92	l[φθk	l[φθk	X
ejde-942	142	93	]	]	X
ejde-942	142	94	(	(	PUNCT
ejde-942	142	95	x	x	SYM
ejde-942	142	96	(	(	PUNCT
ejde-942	142	97	j	j	NOUN
ejde-942	142	98	)	)	PUNCT
ejde-942	142	99	r	r	NOUN
ejde-942	142	100	)	)	PUNCT
ejde-942	142	101	−	−	PROPN
ejde-942	143	1	λkφθk(x	λkφθk(x	INTJ
ejde-942	143	2	(	(	PUNCT
ejde-942	143	3	j	j	NOUN
ejde-942	143	4	)	)	PUNCT
ejde-942	143	5	r	r	NOUN
ejde-942	143	6	)	)	PUNCT
ejde-942	143	7	)	)	PUNCT
ejde-942	143	8	2	2	NUM
ejde-942	143	9	,	,	PUNCT
ejde-942	143	10	(	(	PUNCT
ejde-942	143	11	2.11	2.11	NUM
ejde-942	143	12	)	)	PUNCT
ejde-942	143	13	where	where	SCONJ
ejde-942	143	14	{	{	PUNCT
ejde-942	143	15	x(j)r	x(j)r	PROPN
ejde-942	143	16	}	}	PUNCT
ejde-942	143	17	are	be	AUX
ejde-942	143	18	sets	set	NOUN
ejde-942	143	19	of	of	ADP
ejde-942	143	20	inner	inner	ADJ
ejde-942	143	21	points	point	NOUN
ejde-942	143	22	sampled	sample	VERB
ejde-942	143	23	uniformly	uniformly	ADV
ejde-942	143	24	in	in	ADP
ejde-942	143	25	the	the	DET
ejde-942	143	26	entire	entire	ADJ
ejde-942	143	27	domain	domain	NOUN
ejde-942	143	28	ω	ω	NOUN
ejde-942	143	29	.	.	PUNCT
ejde-942	144	1	the	the	DET
ejde-942	144	2	weight	weight	NOUN
ejde-942	144	3	norm	norm	NOUN
ejde-942	144	4	∥	∥	X
ejde-942	144	5	·	·	PUNCT
ejde-942	144	6	∥σ	∥σ	NOUN
ejde-942	144	7	and	and	CCONJ
ejde-942	144	8	weight	weight	NOUN
ejde-942	144	9	inner	inner	ADJ
ejde-942	144	10	product	product	NOUN
ejde-942	144	11	in	in	ADP
ejde-942	144	12	jnor	jnor	NOUN
ejde-942	144	13	and	and	CCONJ
ejde-942	144	14	jorth	jorth	NOUN
ejde-942	144	15	are	be	AUX
ejde-942	144	16	calculated	calculate	VERB
ejde-942	144	17	by	by	ADP
ejde-942	144	18	the	the	DET
ejde-942	144	19	mid	mid	ADJ
ejde-942	144	20	-	-	ADJ
ejde-942	144	21	point	point	ADJ
ejde-942	144	22	integral	integral	ADJ
ejde-942	144	23	method	method	NOUN
ejde-942	144	24	.	.	PUNCT
ejde-942	145	1	indeed	indeed	ADV
ejde-942	145	2	,	,	PUNCT
ejde-942	145	3	our	our	PRON
ejde-942	145	4	framework	framework	NOUN
ejde-942	145	5	is	be	AUX
ejde-942	145	6	applicable	applicable	ADJ
ejde-942	145	7	not	not	PART
ejde-942	145	8	only	only	ADV
ejde-942	145	9	to	to	ADP
ejde-942	145	10	1	1	NUM
ejde-942	145	11	-	-	PUNCT
ejde-942	145	12	dimensional	dimensional	ADJ
ejde-942	145	13	cases	case	NOUN
ejde-942	145	14	but	but	CCONJ
ejde-942	145	15	also	also	ADV
ejde-942	145	16	to	to	ADP
ejde-942	145	17	2	2	NUM
ejde-942	145	18	-	-	PUNCT
ejde-942	145	19	dimensional	dimensional	ADJ
ejde-942	145	20	cases	case	NOUN
ejde-942	145	21	,	,	PUNCT
ejde-942	145	22	as	as	ADV
ejde-942	145	23	well	well	ADV
ejde-942	145	24	as	as	ADP
ejde-942	145	25	periodic	periodic	ADJ
ejde-942	145	26	scenarios	scenario	NOUN
ejde-942	145	27	.	.	PUNCT
ejde-942	146	1	for	for	ADP
ejde-942	146	2	the	the	DET
ejde-942	146	3	1d	1d	NUM
ejde-942	146	4	case	case	NOUN
ejde-942	146	5	,	,	PUNCT
ejde-942	146	6	the	the	DET
ejde-942	146	7	weight	weight	NOUN
ejde-942	146	8	inner	inner	ADJ
ejde-942	146	9	product	product	NOUN
ejde-942	146	10	is	be	AUX
ejde-942	146	11	calculated	calculate	VERB
ejde-942	146	12	by	by	ADP
ejde-942	146	13	the	the	DET
ejde-942	146	14	mid	mid	ADJ
ejde-942	146	15	-	-	ADJ
ejde-942	146	16	point	point	ADJ
ejde-942	146	17	integral	integral	ADJ
ejde-942	146	18	method	method	NOUN
ejde-942	146	19	:	:	PUNCT
ejde-942	146	20	(	(	PUNCT
ejde-942	146	21	φθk	φθk	INTJ
ejde-942	146	22	,	,	PUNCT
ejde-942	146	23	φθj	φθj	NOUN
ejde-942	146	24	)	)	PUNCT
ejde-942	146	25	σ	σ	NOUN
ejde-942	146	26	=	=	PUNCT
ejde-942	146	27	h	h	PROPN
ejde-942	146	28	n−1∑	n−1∑	PROPN
ejde-942	146	29	m=0	m=0	PROPN
ejde-942	146	30	σ(x	σ(x	PROPN
ejde-942	146	31	(	(	PUNCT
ejde-942	146	32	m	m	NOUN
ejde-942	146	33	)	)	PUNCT
ejde-942	146	34	h	h	NOUN
ejde-942	146	35	)	)	PUNCT
ejde-942	146	36	φθk(x	φθk(x	PROPN
ejde-942	146	37	(	(	PUNCT
ejde-942	146	38	m	m	NOUN
ejde-942	146	39	)	)	PUNCT
ejde-942	146	40	h	h	NOUN
ejde-942	146	41	)	)	PUNCT
ejde-942	146	42	φθj	φθj	ADP
ejde-942	146	43	(	(	PUNCT
ejde-942	146	44	x	x	X
ejde-942	146	45	(	(	PUNCT
ejde-942	146	46	m	m	NOUN
ejde-942	146	47	)	)	PUNCT
ejde-942	146	48	h	h	NOUN
ejde-942	146	49	)	)	PUNCT
ejde-942	146	50	(	(	PUNCT
ejde-942	146	51	2.12	2.12	NUM
ejde-942	146	52	)	)	PUNCT
ejde-942	146	53	where	where	SCONJ
ejde-942	146	54	{	{	PUNCT
ejde-942	146	55	x(m	x(m	PROPN
ejde-942	147	1	)	)	PUNCT
ejde-942	147	2	h	h	PROPN
ejde-942	148	1	|x(m	|x(m	ADJ
ejde-942	148	2	)	)	PUNCT
ejde-942	148	3	h	h	NOUN
ejde-942	148	4	=	=	PUNCT
ejde-942	148	5	h/2	h/2	X
ejde-942	148	6	+	+	CCONJ
ejde-942	148	7	m	m	PROPN
ejde-942	148	8	·	·	PUNCT
ejde-942	148	9	h	h	X
ejde-942	148	10	}	}	PUNCT
ejde-942	148	11	are	be	AUX
ejde-942	148	12	sets	set	NOUN
ejde-942	148	13	of	of	ADP
ejde-942	148	14	integral	integral	ADJ
ejde-942	148	15	points	point	NOUN
ejde-942	148	16	with	with	ADP
ejde-942	148	17	h	h	NOUN
ejde-942	148	18	=	=	SYM
ejde-942	148	19	π	π	PROPN
ejde-942	148	20	/	/	SYM
ejde-942	148	21	n	n	PROPN
ejde-942	148	22	.	.	PUNCT
ejde-942	149	1	the	the	DET
ejde-942	149	2	boundary	boundary	ADJ
ejde-942	149	3	value	value	NOUN
ejde-942	149	4	conditions	condition	NOUN
ejde-942	149	5	for	for	ADP
ejde-942	149	6	the	the	DET
ejde-942	149	7	eigenfunction	eigenfunction	NOUN
ejde-942	149	8	φ	φ	PROPN
ejde-942	149	9	are	be	AUX
ejde-942	149	10	defined	define	VERB
ejde-942	149	11	by	by	ADP
ejde-942	149	12	b[φ	b[φ	NOUN
ejde-942	149	13	]	]	X
ejde-942	149	14	=	=	PUNCT
ejde-942	150	1	[	[	X
ejde-942	150	2	a1φ	a1φ	NOUN
ejde-942	150	3	′(0)−	′(0)−	X
ejde-942	150	4	b1φ(0	b1φ(0	PROPN
ejde-942	150	5	)	)	PUNCT
ejde-942	150	6	,	,	PUNCT
ejde-942	150	7	a2φ	a2φ	VERB
ejde-942	150	8	′(π	′(π	NOUN
ejde-942	150	9	)	)	PUNCT
ejde-942	150	10	+	+	CCONJ
ejde-942	150	11	b2φ(π	b2φ(π	NOUN
ejde-942	150	12	)	)	PUNCT
ejde-942	150	13	]	]	PUNCT
ejde-942	150	14	.	.	PUNCT
ejde-942	151	1	(	(	PUNCT
ejde-942	151	2	2.13	2.13	NUM
ejde-942	151	3	)	)	PUNCT
ejde-942	151	4	in	in	ADP
ejde-942	151	5	the	the	DET
ejde-942	151	6	experiments	experiment	NOUN
ejde-942	151	7	,	,	PUNCT
ejde-942	151	8	the	the	DET
ejde-942	151	9	eigenfunction	eigenfunction	NOUN
ejde-942	151	10	network	network	NOUN
ejde-942	151	11	φθk	φθk	NOUN
ejde-942	151	12	is	be	AUX
ejde-942	151	13	set	set	VERB
ejde-942	151	14	as	as	ADP
ejde-942	151	15	a	a	DET
ejde-942	151	16	fully	fully	ADV
ejde-942	151	17	connected	connected	ADJ
ejde-942	151	18	neural	neural	ADJ
ejde-942	151	19	network	network	NOUN
ejde-942	151	20	with	with	ADP
ejde-942	151	21	4	4	NUM
ejde-942	151	22	hidden	hidden	ADJ
ejde-942	151	23	layers	layer	NOUN
ejde-942	151	24	,	,	PUNCT
ejde-942	151	25	each	each	DET
ejde-942	151	26	consisting	consist	VERB
ejde-942	151	27	of	of	ADP
ejde-942	151	28	32	32	NUM
ejde-942	151	29	neurons	neuron	NOUN
ejde-942	151	30	.	.	PUNCT
ejde-942	152	1	silu	silu	PROPN
ejde-942	152	2	is	be	AUX
ejde-942	152	3	used	use	VERB
ejde-942	152	4	as	as	ADP
ejde-942	152	5	the	the	DET
ejde-942	152	6	activation	activation	NOUN
ejde-942	152	7	function	function	NOUN
ejde-942	152	8	.	.	PUNCT
ejde-942	153	1	during	during	ADP
ejde-942	153	2	training	training	NOUN
ejde-942	153	3	,	,	PUNCT
ejde-942	153	4	we	we	PRON
ejde-942	153	5	utilize	utilize	VERB
ejde-942	153	6	the	the	DET
ejde-942	153	7	adam	adam	PROPN
ejde-942	153	8	optimizer	optimizer	NOUN
ejde-942	153	9	to	to	PART
ejde-942	153	10	update	update	VERB
ejde-942	153	11	the	the	DET
ejde-942	153	12	parameters	parameter	NOUN
ejde-942	153	13	of	of	ADP
ejde-942	153	14	the	the	DET
ejde-942	153	15	neural	neural	ADJ
ejde-942	153	16	network	network	NOUN
ejde-942	153	17	,	,	PUNCT
ejde-942	153	18	with	with	ADP
ejde-942	153	19	the	the	DET
ejde-942	153	20	initial	initial	ADJ
ejde-942	153	21	learning	learning	NOUN
ejde-942	153	22	rate	rate	NOUN
ejde-942	153	23	of	of	ADP
ejde-942	153	24	0.01	0.01	NUM
ejde-942	153	25	,	,	PUNCT
ejde-942	153	26	ejde-2024/53	ejde-2024/53	ADP
ejde-942	153	27	deep	deep	ADJ
ejde-942	153	28	learning	learning	NOUN
ejde-942	153	29	method	method	NOUN
ejde-942	153	30	for	for	ADP
ejde-942	153	31	finding	find	VERB
ejde-942	153	32	eigenpairs	eigenpair	NOUN
ejde-942	153	33	7	7	NUM
ejde-942	153	34	which	which	PRON
ejde-942	153	35	drops	drop	VERB
ejde-942	153	36	by	by	ADP
ejde-942	153	37	10	10	NUM
ejde-942	153	38	%	%	NOUN
ejde-942	153	39	every	every	DET
ejde-942	153	40	1000	1000	NUM
ejde-942	153	41	epoch	epoch	NOUN
ejde-942	153	42	.	.	PUNCT
ejde-942	154	1	a	a	DET
ejde-942	154	2	total	total	NOUN
ejde-942	154	3	of	of	ADP
ejde-942	154	4	50	50	NUM
ejde-942	154	5	,	,	PUNCT
ejde-942	154	6	000	000	NUM
ejde-942	154	7	epochs	epoch	NOUN
ejde-942	154	8	are	be	AUX
ejde-942	154	9	set	set	VERB
ejde-942	154	10	,	,	PUNCT
ejde-942	154	11	ensuring	ensure	VERB
ejde-942	154	12	sufficient	sufficient	ADJ
ejde-942	154	13	time	time	NOUN
ejde-942	154	14	for	for	ADP
ejde-942	154	15	network	network	NOUN
ejde-942	154	16	convergence	convergence	NOUN
ejde-942	154	17	.	.	PUNCT
ejde-942	155	1	this	this	DET
ejde-942	155	2	computation	computation	NOUN
ejde-942	155	3	is	be	AUX
ejde-942	155	4	performed	perform	VERB
ejde-942	155	5	using	use	VERB
ejde-942	155	6	the	the	DET
ejde-942	155	7	backpropagation	backpropagation	NOUN
ejde-942	155	8	algorithm	algorithm	NOUN
ejde-942	155	9	implemented	implement	VERB
ejde-942	155	10	in	in	ADP
ejde-942	155	11	pytorch	pytorch	NOUN
ejde-942	155	12	,	,	PUNCT
ejde-942	155	13	a	a	DET
ejde-942	155	14	deep	deep	ADJ
ejde-942	155	15	learning	learning	NOUN
ejde-942	155	16	framework	framework	NOUN
ejde-942	155	17	that	that	PRON
ejde-942	155	18	facilitates	facilitate	VERB
ejde-942	155	19	automatic	automatic	ADJ
ejde-942	155	20	gradient	gradient	ADJ
ejde-942	155	21	computation	computation	NOUN
ejde-942	155	22	.	.	PUNCT
ejde-942	156	1	3	3	X
ejde-942	156	2	.	.	X
ejde-942	156	3	quantitative	quantitative	ADJ
ejde-942	156	4	estimation	estimation	NOUN
ejde-942	156	5	of	of	ADP
ejde-942	156	6	eigenpairs	eigenpair	NOUN
ejde-942	156	7	we	we	PRON
ejde-942	156	8	consider	consider	VERB
ejde-942	156	9	the	the	DET
ejde-942	156	10	sturm	sturm	NOUN
ejde-942	156	11	-	-	PUNCT
ejde-942	156	12	liouville	liouville	NOUN
ejde-942	156	13	eigenvalue	eigenvalue	NOUN
ejde-942	156	14	problem	problem	NOUN
ejde-942	156	15	:	:	PUNCT
ejde-942	156	16	(	(	PUNCT
ejde-942	156	17	ρ(x)φ′(x))′	ρ(x)φ′(x))′	PROPN
ejde-942	156	18	+	+	CCONJ
ejde-942	156	19	λρ(x)φ(x	λρ(x)φ(x	NOUN
ejde-942	156	20	)	)	PUNCT
ejde-942	156	21	=	=	SYM
ejde-942	156	22	0	0	NUM
ejde-942	156	23	,	,	PUNCT
ejde-942	156	24	(	(	PUNCT
ejde-942	156	25	3.1	3.1	NUM
ejde-942	156	26	)	)	PUNCT
ejde-942	156	27	with	with	ADP
ejde-942	156	28	homogeneous	homogeneous	ADJ
ejde-942	156	29	boundary	boundary	ADJ
ejde-942	156	30	value	value	NOUN
ejde-942	156	31	conditions	condition	NOUN
ejde-942	156	32	a1φ	a1φ	ADJ
ejde-942	156	33	′(0)−	′(0)−	PUNCT
ejde-942	156	34	b1φ(0	b1φ(0	PROPN
ejde-942	156	35	)	)	PUNCT
ejde-942	157	1	=	=	SYM
ejde-942	157	2	0	0	NUM
ejde-942	157	3	,	,	PUNCT
ejde-942	157	4	a2φ	a2φ	ADP
ejde-942	157	5	′(π	′(π	NOUN
ejde-942	157	6	)	)	PUNCT
ejde-942	157	7	+	+	CCONJ
ejde-942	157	8	b2φ(π	b2φ(π	NOUN
ejde-942	157	9	)	)	PUNCT
ejde-942	158	1	=	=	SYM
ejde-942	158	2	0	0	NUM
ejde-942	158	3	,	,	PUNCT
ejde-942	158	4	a2i	a2i	PROPN
ejde-942	158	5	+	+	CCONJ
ejde-942	158	6	b2i	b2i	PROPN
ejde-942	158	7	̸=	̸=	PROPN
ejde-942	158	8	0	0	NUM
ejde-942	158	9	,	,	PUNCT
ejde-942	158	10	i	i	PRON
ejde-942	158	11	=	=	NOUN
ejde-942	158	12	1	1	NUM
ejde-942	158	13	,	,	PUNCT
ejde-942	158	14	2	2	NUM
ejde-942	158	15	.	.	PUNCT
ejde-942	158	16	(	(	PUNCT
ejde-942	158	17	3.2	3.2	NUM
ejde-942	158	18	)	)	PUNCT
ejde-942	159	1	the	the	DET
ejde-942	159	2	boundary	boundary	ADJ
ejde-942	159	3	value	value	NOUN
ejde-942	159	4	conditions	condition	NOUN
ejde-942	159	5	are	be	AUX
ejde-942	159	6	divided	divide	VERB
ejde-942	159	7	into	into	ADP
ejde-942	159	8	6	6	NUM
ejde-942	159	9	cases	case	NOUN
ejde-942	159	10	:	:	PUNCT
ejde-942	159	11	(	(	PUNCT
ejde-942	159	12	1	1	X
ejde-942	159	13	)	)	PUNCT
ejde-942	159	14	dirichlet	dirichlet	PROPN
ejde-942	159	15	boundary	boundary	ADJ
ejde-942	159	16	value	value	NOUN
ejde-942	159	17	condition	condition	NOUN
ejde-942	159	18	:	:	PUNCT
ejde-942	159	19	a1	a1	NOUN
ejde-942	159	20	=	=	SYM
ejde-942	159	21	0	0	NUM
ejde-942	159	22	,	,	PUNCT
ejde-942	159	23	b1	b1	NOUN
ejde-942	159	24	>	>	X
ejde-942	159	25	0,a2	0,a2	NOUN
ejde-942	160	1	=	=	SYM
ejde-942	160	2	0	0	NUM
ejde-942	160	3	,	,	PUNCT
ejde-942	160	4	b2	b2	NOUN
ejde-942	160	5	>	>	X
ejde-942	160	6	0	0	NUM
ejde-942	160	7	;	;	PUNCT
ejde-942	160	8	(	(	PUNCT
ejde-942	160	9	2	2	X
ejde-942	160	10	)	)	PUNCT
ejde-942	160	11	neumann	neumann	PROPN
ejde-942	160	12	boundary	boundary	ADJ
ejde-942	160	13	value	value	NOUN
ejde-942	160	14	condition	condition	NOUN
ejde-942	160	15	:	:	PUNCT
ejde-942	160	16	a1	a1	NOUN
ejde-942	160	17	>	>	X
ejde-942	160	18	0	0	PROPN
ejde-942	160	19	,	,	PUNCT
ejde-942	160	20	b1	b1	NOUN
ejde-942	160	21	=	=	SYM
ejde-942	160	22	0	0	NUM
ejde-942	160	23	,	,	PUNCT
ejde-942	160	24	a2	a2	PROPN
ejde-942	160	25	>	>	X
ejde-942	160	26	0	0	PROPN
ejde-942	160	27	,	,	PUNCT
ejde-942	160	28	b2	b2	NOUN
ejde-942	160	29	=	=	SYM
ejde-942	160	30	0	0	NUM
ejde-942	160	31	;	;	PUNCT
ejde-942	160	32	(	(	PUNCT
ejde-942	160	33	3	3	X
ejde-942	160	34	)	)	PUNCT
ejde-942	160	35	dirichlet	dirichlet	PROPN
ejde-942	160	36	-	-	PUNCT
ejde-942	160	37	neumann	neumann	PROPN
ejde-942	160	38	boundary	boundary	ADJ
ejde-942	160	39	value	value	NOUN
ejde-942	160	40	condition	condition	NOUN
ejde-942	160	41	:	:	PUNCT
ejde-942	161	1	a1	a1	NOUN
ejde-942	161	2	=	=	SYM
ejde-942	161	3	0	0	NUM
ejde-942	161	4	,	,	PUNCT
ejde-942	161	5	b1	b1	VERB
ejde-942	161	6	>	>	X
ejde-942	161	7	0	0	PROPN
ejde-942	161	8	,	,	PUNCT
ejde-942	161	9	a2	a2	PROPN
ejde-942	161	10	>	>	X
ejde-942	161	11	0	0	PROPN
ejde-942	161	12	,	,	PUNCT
ejde-942	161	13	b2	b2	NOUN
ejde-942	161	14	=	=	SYM
ejde-942	161	15	0	0	NUM
ejde-942	161	16	or	or	CCONJ
ejde-942	161	17	a1	a1	VERB
ejde-942	161	18	>	>	X
ejde-942	161	19	0	0	PROPN
ejde-942	161	20	,	,	PUNCT
ejde-942	161	21	b1	b1	NOUN
ejde-942	161	22	=	=	SYM
ejde-942	161	23	0	0	NUM
ejde-942	161	24	,	,	PUNCT
ejde-942	161	25	a2	a2	PROPN
ejde-942	161	26	=	=	SYM
ejde-942	161	27	0	0	NUM
ejde-942	161	28	,	,	PUNCT
ejde-942	161	29	b2	b2	NOUN
ejde-942	161	30	>	>	X
ejde-942	161	31	0	0	NUM
ejde-942	161	32	;	;	PUNCT
ejde-942	161	33	(	(	PUNCT
ejde-942	161	34	4	4	X
ejde-942	161	35	)	)	PUNCT
ejde-942	161	36	mixed	mixed	ADJ
ejde-942	161	37	boundary	boundary	ADJ
ejde-942	161	38	value	value	NOUN
ejde-942	161	39	conditions	condition	NOUN
ejde-942	162	1	i	i	PRON
ejde-942	162	2	:	:	PUNCT
ejde-942	162	3	a1	a1	PROPN
ejde-942	162	4	=	=	SYM
ejde-942	162	5	0	0	NUM
ejde-942	162	6	,	,	PUNCT
ejde-942	162	7	b1	b1	VERB
ejde-942	162	8	>	>	X
ejde-942	162	9	0	0	PROPN
ejde-942	162	10	,	,	PUNCT
ejde-942	162	11	a2	a2	PROPN
ejde-942	162	12	>	>	X
ejde-942	162	13	0	0	PROPN
ejde-942	162	14	,	,	PUNCT
ejde-942	162	15	b2	b2	NOUN
ejde-942	162	16	̸=	̸=	PROPN
ejde-942	162	17	0	0	NUM
ejde-942	162	18	or	or	CCONJ
ejde-942	162	19	a1	a1	VERB
ejde-942	162	20	>	>	X
ejde-942	162	21	0	0	PROPN
ejde-942	162	22	,	,	PUNCT
ejde-942	162	23	b1	b1	NOUN
ejde-942	162	24	̸=	̸=	PROPN
ejde-942	162	25	0	0	NUM
ejde-942	162	26	,	,	PUNCT
ejde-942	162	27	a2	a2	PROPN
ejde-942	162	28	=	=	SYM
ejde-942	162	29	0	0	NUM
ejde-942	162	30	,	,	PUNCT
ejde-942	162	31	b2	b2	NOUN
ejde-942	162	32	>	>	X
ejde-942	162	33	0	0	NUM
ejde-942	162	34	;	;	PUNCT
ejde-942	162	35	(	(	PUNCT
ejde-942	162	36	5	5	X
ejde-942	162	37	)	)	PUNCT
ejde-942	162	38	mixed	mixed	ADJ
ejde-942	162	39	boundary	boundary	ADJ
ejde-942	162	40	value	value	NOUN
ejde-942	162	41	conditions	condition	NOUN
ejde-942	162	42	ii	ii	NOUN
ejde-942	162	43	:	:	PUNCT
ejde-942	162	44	a1	a1	VERB
ejde-942	162	45	>	>	X
ejde-942	162	46	0	0	PROPN
ejde-942	162	47	,	,	PUNCT
ejde-942	162	48	b1	b1	NOUN
ejde-942	162	49	=	=	SYM
ejde-942	162	50	0	0	NUM
ejde-942	162	51	,	,	PUNCT
ejde-942	162	52	a2	a2	PROPN
ejde-942	162	53	>	>	X
ejde-942	162	54	0	0	PROPN
ejde-942	162	55	,	,	PUNCT
ejde-942	162	56	b2	b2	NOUN
ejde-942	162	57	̸=	̸=	PROPN
ejde-942	162	58	0	0	NUM
ejde-942	162	59	or	or	CCONJ
ejde-942	162	60	a1	a1	VERB
ejde-942	162	61	>	>	X
ejde-942	162	62	0	0	PROPN
ejde-942	162	63	,	,	PUNCT
ejde-942	162	64	b1	b1	NOUN
ejde-942	162	65	̸=	̸=	PROPN
ejde-942	162	66	0	0	NUM
ejde-942	162	67	,	,	PUNCT
ejde-942	162	68	a2	a2	PROPN
ejde-942	162	69	>	>	X
ejde-942	162	70	0	0	PROPN
ejde-942	162	71	,	,	PUNCT
ejde-942	162	72	b2	b2	NOUN
ejde-942	162	73	=	=	SYM
ejde-942	162	74	0	0	NUM
ejde-942	162	75	;	;	PUNCT
ejde-942	162	76	(	(	PUNCT
ejde-942	162	77	6	6	X
ejde-942	162	78	)	)	PUNCT
ejde-942	162	79	general	general	ADJ
ejde-942	162	80	boundary	boundary	ADJ
ejde-942	162	81	value	value	NOUN
ejde-942	162	82	condition	condition	NOUN
ejde-942	162	83	:	:	PUNCT
ejde-942	162	84	a1	a1	NOUN
ejde-942	162	85	>	>	X
ejde-942	162	86	0	0	PROPN
ejde-942	162	87	,	,	PUNCT
ejde-942	162	88	b1	b1	NOUN
ejde-942	162	89	̸=	̸=	PROPN
ejde-942	162	90	0	0	NUM
ejde-942	162	91	,	,	PUNCT
ejde-942	162	92	a2	a2	PROPN
ejde-942	162	93	>	>	X
ejde-942	162	94	0	0	PROPN
ejde-942	162	95	,	,	PUNCT
ejde-942	162	96	b2	b2	NOUN
ejde-942	162	97	̸=	̸=	PROPN
ejde-942	162	98	0	0	NUM
ejde-942	162	99	.	.	PUNCT
ejde-942	162	100	assume	assume	VERB
ejde-942	162	101	that	that	SCONJ
ejde-942	162	102	ρ(x	ρ(x	NOUN
ejde-942	162	103	)	)	PUNCT
ejde-942	162	104	∈	∈	PROPN
ejde-942	162	105	h2((0	h2((0	PROPN
ejde-942	162	106	,	,	PUNCT
ejde-942	162	107	π),r+	π),r+	PROPN
ejde-942	162	108	)	)	PUNCT
ejde-942	162	109	.	.	PUNCT
ejde-942	163	1	let	let	VERB
ejde-942	163	2	z(x	z(x	NUM
ejde-942	163	3	)	)	PUNCT
ejde-942	163	4	=	=	SYM
ejde-942	163	5	√	√	NUM
ejde-942	163	6	ρ(x)φ(x	ρ(x)φ(x	NOUN
ejde-942	163	7	)	)	PUNCT
ejde-942	163	8	,	,	PUNCT
ejde-942	163	9	then	then	ADV
ejde-942	163	10	the	the	DET
ejde-942	163	11	above	above	ADJ
ejde-942	163	12	equation	equation	NOUN
ejde-942	163	13	can	can	AUX
ejde-942	163	14	be	be	AUX
ejde-942	163	15	transformed	transform	VERB
ejde-942	163	16	to	to	ADP
ejde-942	163	17	z′′(x	z′′(x	NOUN
ejde-942	163	18	)	)	PUNCT
ejde-942	164	1	+	+	CCONJ
ejde-942	164	2	(	(	PUNCT
ejde-942	164	3	λ−	λ−	PROPN
ejde-942	164	4	ηρ(x))z(x	ηρ(x))z(x	ADJ
ejde-942	164	5	)	)	PUNCT
ejde-942	164	6	=	=	SYM
ejde-942	164	7	0	0	NUM
ejde-942	164	8	,	,	PUNCT
ejde-942	164	9	where	where	SCONJ
ejde-942	164	10	ηρ(x	ηρ(x	PUNCT
ejde-942	164	11	)	)	PUNCT
ejde-942	164	12	=	=	SYM
ejde-942	164	13	1	1	NUM
ejde-942	164	14	2	2	NUM
ejde-942	164	15	ρ′′	ρ′′	NOUN
ejde-942	164	16	ρ	ρ	NOUN
ejde-942	164	17	−	−	PROPN
ejde-942	164	18	1	1	NUM
ejde-942	164	19	4	4	NUM
ejde-942	164	20	(	(	PUNCT
ejde-942	164	21	ρ′	ρ′	NOUN
ejde-942	164	22	ρ	ρ	NOUN
ejde-942	164	23	)	)	PUNCT
ejde-942	164	24	2	2	NUM
ejde-942	164	25	is	be	AUX
ejde-942	164	26	assumed	assume	VERB
ejde-942	164	27	to	to	PART
ejde-942	164	28	be	be	AUX
ejde-942	164	29	a	a	DET
ejde-942	164	30	real	real	ADJ
ejde-942	164	31	function	function	NOUN
ejde-942	164	32	in	in	ADP
ejde-942	164	33	l2(0	l2(0	PROPN
ejde-942	164	34	,	,	PUNCT
ejde-942	164	35	π	π	PROPN
ejde-942	164	36	)	)	PUNCT
ejde-942	164	37	.	.	PUNCT
ejde-942	165	1	the	the	DET
ejde-942	165	2	boundary	boundary	ADJ
ejde-942	165	3	value	value	NOUN
ejde-942	165	4	condition	condition	NOUN
ejde-942	165	5	is	be	AUX
ejde-942	165	6	transformed	transform	VERB
ejde-942	165	7	to	to	ADP
ejde-942	165	8	a	a	DET
ejde-942	165	9	new	new	ADJ
ejde-942	165	10	one	one	NOUN
ejde-942	165	11	:	:	PUNCT
ejde-942	165	12	a1z	a1z	PROPN
ejde-942	165	13	′(0)−	′(0)−	X
ejde-942	165	14	(	(	PUNCT
ejde-942	165	15	a1	a1	NOUN
ejde-942	165	16	2	2	NUM
ejde-942	165	17	ρ′(0	ρ′(0	NOUN
ejde-942	165	18	)	)	PUNCT
ejde-942	165	19	ρ(0	ρ(0	PROPN
ejde-942	165	20	)	)	PUNCT
ejde-942	166	1	+	+	NUM
ejde-942	166	2	b1	b1	NOUN
ejde-942	166	3	)	)	PUNCT
ejde-942	166	4	z(0	z(0	ADV
ejde-942	166	5	)	)	PUNCT
ejde-942	166	6	=	=	SYM
ejde-942	166	7	0	0	NUM
ejde-942	166	8	,	,	PUNCT
ejde-942	166	9	a2z	a2z	PROPN
ejde-942	166	10	′(π	′(π	NOUN
ejde-942	166	11	)	)	PUNCT
ejde-942	166	12	+	+	CCONJ
ejde-942	166	13	(	(	PUNCT
ejde-942	166	14	b2	b2	NOUN
ejde-942	166	15	−	−	PROPN
ejde-942	166	16	a2	a2	PROPN
ejde-942	166	17	2	2	NUM
ejde-942	166	18	ρ′(π	ρ′(π	NOUN
ejde-942	166	19	)	)	PUNCT
ejde-942	166	20	ρ(π	ρ(π	PROPN
ejde-942	166	21	)	)	PUNCT
ejde-942	166	22	)	)	PUNCT
ejde-942	166	23	z(π	z(π	PROPN
ejde-942	166	24	)	)	PUNCT
ejde-942	166	25	=	=	SYM
ejde-942	167	1	0	0	X
ejde-942	167	2	.	.	PUNCT
ejde-942	168	1	let	let	AUX
ejde-942	168	2	(	(	PUNCT
ejde-942	168	3	λk	λk	X
ejde-942	168	4	,	,	PUNCT
ejde-942	168	5	φk	φk	AUX
ejde-942	168	6	)	)	PUNCT
ejde-942	168	7	be	be	AUX
ejde-942	168	8	the	the	DET
ejde-942	168	9	exact	exact	ADJ
ejde-942	168	10	eigenpairs	eigenpair	NOUN
ejde-942	168	11	of	of	ADP
ejde-942	168	12	(	(	PUNCT
ejde-942	168	13	3.1)-(3.2	3.1)-(3.2	NUM
ejde-942	168	14	)	)	PUNCT
ejde-942	168	15	.	.	PUNCT
ejde-942	169	1	let	let	AUX
ejde-942	169	2	(	(	PUNCT
ejde-942	169	3	λk	λk	NOUN
ejde-942	169	4	,	,	PUNCT
ejde-942	169	5	φθk	φθk	NOUN
ejde-942	169	6	)	)	PUNCT
ejde-942	169	7	be	be	AUX
ejde-942	169	8	approximate	approximate	ADJ
ejde-942	169	9	eigenpairs	eigenpair	NOUN
ejde-942	169	10	calculated	calculate	VERB
ejde-942	169	11	by	by	ADP
ejde-942	169	12	our	our	PRON
ejde-942	169	13	neural	neural	ADJ
ejde-942	169	14	networks	network	NOUN
ejde-942	169	15	.	.	PUNCT
ejde-942	170	1	denote	denote	VERB
ejde-942	170	2	ρ0	ρ0	X
ejde-942	170	3	:	:	PUNCT
ejde-942	170	4	=	=	SYM
ejde-942	170	5	ess	ess	PROPN
ejde-942	170	6	inf	inf	PROPN
ejde-942	170	7	ηρ(x	ηρ(x	PUNCT
ejde-942	170	8	)	)	PUNCT
ejde-942	170	9	and	and	CCONJ
ejde-942	170	10	ρ1	ρ1	NOUN
ejde-942	170	11	:	:	PUNCT
ejde-942	171	1	=	=	SYM
ejde-942	171	2	ρ0	ρ0	PROPN
ejde-942	171	3	+	+	NOUN
ejde-942	171	4	2	2	NUM
ejde-942	171	5	π	π	NOUN
ejde-942	171	6	∫	∫	PROPN
ejde-942	171	7	π	π	X
ejde-942	171	8	0	0	PUNCT
ejde-942	171	9	{	{	PUNCT
ejde-942	171	10	ηρ(x)−	ηρ(x)−	PROPN
ejde-942	171	11	ρ0}+	ρ0}+	NUM
ejde-942	171	12	dx	dx	PROPN
ejde-942	171	13	,	,	PUNCT
ejde-942	171	14	where	where	SCONJ
ejde-942	171	15	{	{	PUNCT
ejde-942	171	16	h(x)}+	h(x)}+	NOUN
ejde-942	171	17	=	=	SYM
ejde-942	171	18	max{h(x	max{h(x	PROPN
ejde-942	171	19	)	)	PUNCT
ejde-942	171	20	,	,	PUNCT
ejde-942	171	21	0	0	NUM
ejde-942	171	22	}	}	PUNCT
ejde-942	171	23	.	.	PUNCT
ejde-942	172	1	now	now	ADV
ejde-942	172	2	,	,	PUNCT
ejde-942	172	3	we	we	PRON
ejde-942	172	4	give	give	VERB
ejde-942	172	5	the	the	DET
ejde-942	172	6	estimation	estimation	NOUN
ejde-942	172	7	of	of	ADP
ejde-942	172	8	λk	λk	NOUN
ejde-942	172	9	for	for	ADP
ejde-942	172	10	sturmliouville	sturmliouville	NOUN
ejde-942	172	11	eigenvalue	eigenvalue	NOUN
ejde-942	172	12	probems	probem	NOUN
ejde-942	172	13	.	.	PUNCT
ejde-942	173	1	the	the	DET
ejde-942	173	2	eigenvalue	eigenvalue	PROPN
ejde-942	173	3	estimation	estimation	NOUN
ejde-942	173	4	for	for	ADP
ejde-942	173	5	dirichlet	dirichlet	PROPN
ejde-942	173	6	boundary	boundary	ADJ
ejde-942	173	7	value	value	NOUN
ejde-942	173	8	conditions	condition	NOUN
ejde-942	173	9	has	have	AUX
ejde-942	173	10	been	be	AUX
ejde-942	173	11	well	well	ADV
ejde-942	173	12	studied	study	VERB
ejde-942	173	13	.	.	PUNCT
ejde-942	174	1	lemma	lemma	PROPN
ejde-942	174	2	3.1	3.1	NUM
ejde-942	174	3	(	(	PUNCT
ejde-942	174	4	[	[	X
ejde-942	174	5	22	22	NUM
ejde-942	174	6	]	]	PUNCT
ejde-942	174	7	)	)	PUNCT
ejde-942	174	8	.	.	PUNCT
ejde-942	175	1	the	the	DET
ejde-942	175	2	sturm	sturm	NOUN
ejde-942	175	3	-	-	PUNCT
ejde-942	175	4	liouville	liouville	NOUN
ejde-942	175	5	problems	problem	NOUN
ejde-942	175	6	(	(	PUNCT
ejde-942	175	7	3.1	3.1	NUM
ejde-942	175	8	)	)	PUNCT
ejde-942	175	9	with	with	ADP
ejde-942	175	10	case	case	NOUN
ejde-942	175	11	(	(	PUNCT
ejde-942	175	12	1	1	X
ejde-942	175	13	)	)	PUNCT
ejde-942	175	14	have	have	VERB
ejde-942	175	15	the	the	DET
ejde-942	175	16	following	follow	VERB
ejde-942	175	17	inequalities	inequality	NOUN
ejde-942	175	18	:	:	PUNCT
ejde-942	175	19	k2	k2	PROPN
ejde-942	175	20	+	+	CCONJ
ejde-942	175	21	ρ0	ρ0	PROPN
ejde-942	175	22	≤	≤	PROPN
ejde-942	175	23	λk	λk	ADP
ejde-942	175	24	≤	≤	PROPN
ejde-942	175	25	k2	k2	PROPN
ejde-942	175	26	+	+	CCONJ
ejde-942	175	27	ρ1	ρ1	PROPN
ejde-942	175	28	,	,	PUNCT
ejde-942	175	29	k	k	PROPN
ejde-942	175	30	∈	∈	PROPN
ejde-942	175	31	n+	n+	PROPN
ejde-942	175	32	.	.	PROPN
ejde-942	175	33	8	8	NUM
ejde-942	175	34	s.	s.	PROPN
ejde-942	175	35	zhang	zhang	PROPN
ejde-942	175	36	,	,	PUNCT
ejde-942	175	37	j.	j.	PROPN
ejde-942	175	38	zu	zu	PROPN
ejde-942	175	39	,	,	PUNCT
ejde-942	175	40	j.	j.	PROPN
ejde-942	175	41	zhang	zhang	PROPN
ejde-942	175	42	ejde-2024/53	ejde-2024/53	VERB
ejde-942	175	43	the	the	DET
ejde-942	175	44	proof	proof	NOUN
ejde-942	175	45	of	of	ADP
ejde-942	175	46	above	above	ADP
ejde-942	175	47	lemma	lemma	PROPN
ejde-942	175	48	can	can	AUX
ejde-942	175	49	be	be	AUX
ejde-942	175	50	seen	see	VERB
ejde-942	175	51	in	in	ADP
ejde-942	175	52	[	[	X
ejde-942	175	53	22	22	NUM
ejde-942	175	54	]	]	PUNCT
ejde-942	175	55	.	.	PUNCT
ejde-942	176	1	notice	notice	VERB
ejde-942	176	2	that	that	SCONJ
ejde-942	176	3	we	we	PRON
ejde-942	176	4	have	have	AUX
ejde-942	176	5	not	not	PART
ejde-942	176	6	assumed	assume	VERB
ejde-942	176	7	that	that	SCONJ
ejde-942	176	8	ρ0	ρ0	PROPN
ejde-942	176	9	>	>	X
ejde-942	176	10	0	0	PROPN
ejde-942	176	11	,	,	PUNCT
ejde-942	176	12	so	so	ADV
ejde-942	176	13	the	the	DET
ejde-942	176	14	case	case	NOUN
ejde-942	176	15	ρ(x	ρ(x	NOUN
ejde-942	176	16	)	)	PUNCT
ejde-942	176	17	≡	≡	PROPN
ejde-942	176	18	constant	constant	ADJ
ejde-942	176	19	is	be	AUX
ejde-942	176	20	included	include	VERB
ejde-942	176	21	.	.	PUNCT
ejde-942	177	1	denote	denote	VERB
ejde-942	177	2	α1	α1	PROPN
ejde-942	177	3	:	:	PUNCT
ejde-942	177	4	=	=	SYM
ejde-942	177	5	ρ′(0	ρ′(0	NOUN
ejde-942	177	6	)	)	PUNCT
ejde-942	177	7	2ρ(0	2ρ(0	NUM
ejde-942	177	8	)	)	PUNCT
ejde-942	178	1	+	+	CCONJ
ejde-942	178	2	b1	b1	NOUN
ejde-942	178	3	a1	a1	NOUN
ejde-942	178	4	and	and	CCONJ
ejde-942	178	5	α2	α2	ADJ
ejde-942	178	6	:	:	PUNCT
ejde-942	178	7	=	=	PUNCT
ejde-942	178	8	−	−	ADP
ejde-942	178	9	ρ′(π	ρ′(π	NUM
ejde-942	178	10	)	)	PUNCT
ejde-942	178	11	2ρ(π	2ρ(π	NUM
ejde-942	178	12	)	)	PUNCT
ejde-942	179	1	+	+	NUM
ejde-942	179	2	b2	b2	NOUN
ejde-942	179	3	a2	a2	PROPN
ejde-942	179	4	.	.	PUNCT
ejde-942	180	1	for	for	ADP
ejde-942	180	2	other	other	ADJ
ejde-942	180	3	boundary	boundary	ADJ
ejde-942	180	4	value	value	NOUN
ejde-942	180	5	problems	problem	NOUN
ejde-942	180	6	,	,	PUNCT
ejde-942	180	7	we	we	PRON
ejde-942	180	8	have	have	VERB
ejde-942	180	9	the	the	DET
ejde-942	180	10	following	follow	VERB
ejde-942	180	11	result	result	NOUN
ejde-942	180	12	.	.	PUNCT
ejde-942	181	1	lemma	lemma	PROPN
ejde-942	181	2	3.2	3.2	NUM
ejde-942	181	3	.	.	PUNCT
ejde-942	182	1	let	let	VERB
ejde-942	182	2	k	k	PROPN
ejde-942	182	3	∈	∈	PROPN
ejde-942	182	4	n+	n+	PROPN
ejde-942	182	5	.	.	PUNCT
ejde-942	183	1	if	if	SCONJ
ejde-942	183	2	a2	a2	PROPN
ejde-942	183	3	=	=	SYM
ejde-942	183	4	0	0	NUM
ejde-942	183	5	and	and	CCONJ
ejde-942	183	6	α1	α1	PROPN
ejde-942	183	7	≥	≥	NUM
ejde-942	183	8	0	0	NUM
ejde-942	183	9	(	(	PUNCT
ejde-942	183	10	resp	resp	NOUN
ejde-942	183	11	.	.	PUNCT
ejde-942	184	1	a1	a1	NOUN
ejde-942	184	2	=	=	NOUN
ejde-942	184	3	0	0	PROPN
ejde-942	184	4	and	and	CCONJ
ejde-942	184	5	α2	α2	ADJ
ejde-942	184	6	≥	≥	NOUN
ejde-942	184	7	0	0	NUM
ejde-942	184	8	)	)	PUNCT
ejde-942	184	9	,	,	PUNCT
ejde-942	184	10	the	the	DET
ejde-942	184	11	sturm	sturm	NOUN
ejde-942	184	12	-	-	PUNCT
ejde-942	184	13	liouville	liouville	NOUN
ejde-942	184	14	problems	problem	NOUN
ejde-942	184	15	(	(	PUNCT
ejde-942	184	16	3.1	3.1	NUM
ejde-942	184	17	)	)	PUNCT
ejde-942	184	18	with	with	ADP
ejde-942	184	19	case	case	NOUN
ejde-942	184	20	(	(	PUNCT
ejde-942	184	21	3	3	NUM
ejde-942	184	22	)	)	PUNCT
ejde-942	184	23	and	and	CCONJ
ejde-942	184	24	(	(	PUNCT
ejde-942	184	25	4	4	X
ejde-942	184	26	)	)	PUNCT
ejde-942	184	27	satisfy	satisfy	NOUN
ejde-942	184	28	(	(	PUNCT
ejde-942	184	29	k	k	NOUN
ejde-942	184	30	−	−	PROPN
ejde-942	185	1	1	1	NUM
ejde-942	185	2	2	2	NUM
ejde-942	185	3	)	)	PUNCT
ejde-942	185	4	2	2	NUM
ejde-942	186	1	+	+	CCONJ
ejde-942	186	2	ρ0	ρ0	PROPN
ejde-942	186	3	≤	≤	PROPN
ejde-942	186	4	λk	λk	ADP
ejde-942	186	5	≤	≤	PROPN
ejde-942	186	6	(	(	PUNCT
ejde-942	186	7	k	k	NOUN
ejde-942	186	8	−	−	PROPN
ejde-942	186	9	1	1	NUM
ejde-942	186	10	2	2	NUM
ejde-942	186	11	)	)	SYM
ejde-942	186	12	2	2	NUM
ejde-942	186	13	+	+	NUM
ejde-942	186	14	ρ1	ρ1	NOUN
ejde-942	186	15	+	+	CCONJ
ejde-942	186	16	2α1	2α1	NUM
ejde-942	186	17	π	π	NOUN
ejde-942	186	18	(	(	PUNCT
ejde-942	186	19	resp	resp	NOUN
ejde-942	186	20	.	.	PUNCT
ejde-942	187	1	≤	≤	NUM
ejde-942	187	2	(	(	PUNCT
ejde-942	187	3	k	k	NOUN
ejde-942	187	4	−	−	PROPN
ejde-942	187	5	1	1	NUM
ejde-942	187	6	2	2	NUM
ejde-942	187	7	)	)	SYM
ejde-942	187	8	2	2	NUM
ejde-942	187	9	+	+	NUM
ejde-942	187	10	ρ1	ρ1	NOUN
ejde-942	187	11	+	+	CCONJ
ejde-942	187	12	2α2	2α2	NUM
ejde-942	187	13	π	π	NOUN
ejde-942	187	14	)	)	PUNCT
ejde-942	187	15	.	.	PUNCT
ejde-942	188	1	proof	proof	NOUN
ejde-942	188	2	.	.	PUNCT
ejde-942	189	1	the	the	DET
ejde-942	189	2	case	case	NOUN
ejde-942	189	3	a2	a2	NOUN
ejde-942	189	4	=	=	SYM
ejde-942	189	5	0	0	NUM
ejde-942	189	6	and	and	CCONJ
ejde-942	189	7	α1	α1	PROPN
ejde-942	189	8	=	=	SYM
ejde-942	189	9	0	0	NUM
ejde-942	189	10	can	can	AUX
ejde-942	189	11	be	be	AUX
ejde-942	189	12	found	find	VERB
ejde-942	189	13	in	in	ADP
ejde-942	189	14	[	[	X
ejde-942	189	15	11	11	NUM
ejde-942	189	16	]	]	PUNCT
ejde-942	189	17	.	.	PUNCT
ejde-942	190	1	now	now	ADV
ejde-942	190	2	we	we	PRON
ejde-942	190	3	prove	prove	VERB
ejde-942	190	4	that	that	SCONJ
ejde-942	190	5	it	it	PRON
ejde-942	190	6	still	still	ADV
ejde-942	190	7	holds	hold	VERB
ejde-942	190	8	for	for	ADP
ejde-942	190	9	the	the	DET
ejde-942	190	10	case	case	NOUN
ejde-942	190	11	a2	a2	NOUN
ejde-942	190	12	=	=	SYM
ejde-942	190	13	0	0	NUM
ejde-942	190	14	and	and	CCONJ
ejde-942	190	15	α1	α1	PROPN
ejde-942	190	16	>	>	X
ejde-942	190	17	0	0	X
ejde-942	190	18	.	.	PUNCT
ejde-942	191	1	it	it	PRON
ejde-942	191	2	is	be	AUX
ejde-942	191	3	obviously	obviously	ADV
ejde-942	191	4	that	that	PRON
ejde-942	191	5	λk	λk	ADP
ejde-942	191	6	>	>	X
ejde-942	191	7	ρ0	ρ0	PROPN
ejde-942	191	8	,	,	PUNCT
ejde-942	191	9	k	k	PROPN
ejde-942	191	10	∈	∈	PROPN
ejde-942	191	11	n+	n+	PROPN
ejde-942	191	12	.	.	PUNCT
ejde-942	192	1	we	we	PRON
ejde-942	192	2	introduce	introduce	VERB
ejde-942	192	3	the	the	DET
ejde-942	192	4	prüfer	prüfer	NOUN
ejde-942	192	5	transformation	transformation	NOUN
ejde-942	192	6	:	:	PUNCT
ejde-942	192	7	zk	zk	X
ejde-942	192	8	=	=	PUNCT
ejde-942	192	9	r	r	NOUN
ejde-942	192	10	sin	sin	NOUN
ejde-942	192	11	θ	θ	PROPN
ejde-942	192	12	,	,	PUNCT
ejde-942	192	13	z′k	z′k	NOUN
ejde-942	192	14	=	=	PUNCT
ejde-942	193	1	√	√	PROPN
ejde-942	193	2	λk	λk	ADP
ejde-942	193	3	−	−	PROPN
ejde-942	194	1	ρ0r	ρ0r	PUNCT
ejde-942	194	2	cos	cos	PROPN
ejde-942	194	3	θ	θ	PROPN
ejde-942	194	4	,	,	PUNCT
ejde-942	194	5	where	where	SCONJ
ejde-942	194	6	r(x	r(x	NOUN
ejde-942	194	7	)	)	PUNCT
ejde-942	194	8	>	>	X
ejde-942	194	9	0	0	X
ejde-942	194	10	.	.	PUNCT
ejde-942	194	11	assume	assume	VERB
ejde-942	194	12	that	that	SCONJ
ejde-942	194	13	zk(x	zk(x	NOUN
ejde-942	194	14	)	)	PUNCT
ejde-942	194	15	has	have	VERB
ejde-942	194	16	(	(	PUNCT
ejde-942	194	17	k	k	NOUN
ejde-942	194	18	−	−	PROPN
ejde-942	194	19	1	1	X
ejde-942	194	20	)	)	PUNCT
ejde-942	194	21	zeros	zero	NOUN
ejde-942	194	22	in	in	ADP
ejde-942	194	23	the	the	DET
ejde-942	194	24	interval	interval	NOUN
ejde-942	194	25	(	(	PUNCT
ejde-942	194	26	0	0	NUM
ejde-942	194	27	,	,	PUNCT
ejde-942	194	28	π	π	NOUN
ejde-942	194	29	)	)	PUNCT
ejde-942	194	30	denoted	denote	VERB
ejde-942	194	31	as	as	ADP
ejde-942	194	32	0	0	NUM
ejde-942	194	33	=	=	SYM
ejde-942	194	34	κ0	κ0	ADP
ejde-942	194	35	<	<	X
ejde-942	194	36	κ1	κ1	NOUN
ejde-942	194	37	<	<	X
ejde-942	194	38	κ2	κ2	PROPN
ejde-942	194	39	<	<	X
ejde-942	194	40	·	·	PUNCT
ejde-942	194	41	·	·	PUNCT
ejde-942	194	42	·	·	PUNCT
ejde-942	195	1	<	<	X
ejde-942	195	2	κk−1	κk−1	NOUN
ejde-942	195	3	<	<	X
ejde-942	195	4	κk	κk	NOUN
ejde-942	195	5	=	=	SYM
ejde-942	195	6	π	π	PROPN
ejde-942	195	7	.	.	PUNCT
ejde-942	195	8	define	define	VERB
ejde-942	195	9	θ(κi	θ(κi	NOUN
ejde-942	195	10	)	)	PUNCT
ejde-942	195	11	=	=	SYM
ejde-942	196	1	iπ	iπ	NOUN
ejde-942	196	2	,	,	PUNCT
ejde-942	196	3	i	i	PRON
ejde-942	196	4	=	=	NOUN
ejde-942	196	5	1	1	NUM
ejde-942	196	6	,	,	PUNCT
ejde-942	196	7	2	2	NUM
ejde-942	196	8	,	,	PUNCT
ejde-942	196	9	.	.	PUNCT
ejde-942	196	10	.	.	PUNCT
ejde-942	196	11	.	.	PUNCT
ejde-942	197	1	,	,	PUNCT
ejde-942	198	1	k	k	PROPN
ejde-942	199	1	−	−	NOUN
ejde-942	199	2	1	1	X
ejde-942	199	3	.	.	PUNCT
ejde-942	199	4	using	use	VERB
ejde-942	199	5	the	the	DET
ejde-942	199	6	boundary	boundary	ADJ
ejde-942	199	7	conditions	condition	NOUN
ejde-942	199	8	,	,	PUNCT
ejde-942	199	9	we	we	PRON
ejde-942	199	10	obtain	obtain	VERB
ejde-942	199	11	θ(0	θ(0	PROPN
ejde-942	199	12	)	)	PUNCT
ejde-942	199	13	=	=	SYM
ejde-942	199	14	arctan	arctan	PROPN
ejde-942	199	15	(	(	PUNCT
ejde-942	199	16	1	1	NUM
ejde-942	199	17	α1	α1	PROPN
ejde-942	199	18	√	√	INTJ
ejde-942	199	19	λk	λk	ADP
ejde-942	199	20	−	−	PROPN
ejde-942	199	21	ρ0	ρ0	PROPN
ejde-942	199	22	)	)	PUNCT
ejde-942	199	23	,	,	PUNCT
ejde-942	199	24	θ(π	θ(π	PROPN
ejde-942	199	25	)	)	PUNCT
ejde-942	199	26	=	=	SYM
ejde-942	199	27	kπ	kπ	PROPN
ejde-942	199	28	.	.	PUNCT
ejde-942	200	1	firstly	firstly	ADV
ejde-942	200	2	,	,	PUNCT
ejde-942	200	3	we	we	PRON
ejde-942	200	4	estimate	estimate	VERB
ejde-942	200	5	the	the	DET
ejde-942	200	6	lower	low	ADJ
ejde-942	200	7	bound	bind	VERB
ejde-942	200	8	of	of	ADP
ejde-942	200	9	λk	λk	PROPN
ejde-942	200	10	.	.	PUNCT
ejde-942	201	1	it	it	PRON
ejde-942	201	2	is	be	AUX
ejde-942	201	3	not	not	PART
ejde-942	201	4	difficult	difficult	ADJ
ejde-942	201	5	to	to	PART
ejde-942	201	6	know	know	VERB
ejde-942	201	7	that	that	DET
ejde-942	201	8	θ′	θ′	NOUN
ejde-942	201	9	=	=	SYM
ejde-942	201	10	√	√	NUM
ejde-942	201	11	λk	λk	ADP
ejde-942	201	12	−	−	PROPN
ejde-942	201	13	ρ0	ρ0	PROPN
ejde-942	201	14	−	−	PROPN
ejde-942	202	1	(	(	PUNCT
ejde-942	202	2	ηu(x)−	ηu(x)−	PROPN
ejde-942	202	3	ρ0	ρ0	PROPN
ejde-942	202	4	)	)	PUNCT
ejde-942	202	5	sin	sin	NOUN
ejde-942	202	6	2	2	NUM
ejde-942	202	7	θ√	θ√	PROPN
ejde-942	202	8	λk	λk	ADP
ejde-942	202	9	−	−	PROPN
ejde-942	202	10	ρ0	ρ0	PROPN
ejde-942	202	11	≤	≤	PROPN
ejde-942	202	12	√	√	ADP
ejde-942	202	13	λk	λk	ADP
ejde-942	202	14	−	−	PROPN
ejde-942	202	15	ρ0	ρ0	PROPN
ejde-942	202	16	.	.	PUNCT
ejde-942	203	1	integrating	integrate	VERB
ejde-942	203	2	both	both	DET
ejde-942	203	3	sides	side	NOUN
ejde-942	203	4	of	of	ADP
ejde-942	203	5	the	the	DET
ejde-942	203	6	inequality	inequality	NOUN
ejde-942	203	7	over	over	ADP
ejde-942	203	8	(	(	PUNCT
ejde-942	203	9	0	0	NUM
ejde-942	203	10	,	,	PUNCT
ejde-942	203	11	π	π	PROPN
ejde-942	203	12	)	)	PUNCT
ejde-942	203	13	,	,	PUNCT
ejde-942	203	14	we	we	PRON
ejde-942	203	15	obtain	obtain	VERB
ejde-942	203	16	θ(π)−	θ(π)−	NUM
ejde-942	203	17	θ(0	θ(0	PROPN
ejde-942	203	18	)	)	PUNCT
ejde-942	203	19	≤	≤	NOUN
ejde-942	204	1	√	√	ADP
ejde-942	204	2	λk	λk	ADP
ejde-942	204	3	−	−	PROPN
ejde-942	204	4	ρ0π	ρ0π	PROPN
ejde-942	204	5	.	.	PUNCT
ejde-942	205	1	since	since	SCONJ
ejde-942	205	2	α1	α1	PROPN
ejde-942	205	3	>	>	X
ejde-942	205	4	0	0	NUM
ejde-942	205	5	,	,	PUNCT
ejde-942	205	6	utilizing	utilize	VERB
ejde-942	205	7	the	the	DET
ejde-942	205	8	properties	property	NOUN
ejde-942	205	9	of	of	ADP
ejde-942	205	10	arctan(x	arctan(x	PROPN
ejde-942	205	11	)	)	PUNCT
ejde-942	205	12	,	,	PUNCT
ejde-942	205	13	we	we	PRON
ejde-942	205	14	know	know	VERB
ejde-942	205	15	that	that	SCONJ
ejde-942	205	16	θ(0	θ(0	PROPN
ejde-942	205	17	)	)	PUNCT
ejde-942	205	18	=	=	SYM
ejde-942	205	19	arctan	arctan	PROPN
ejde-942	205	20	(	(	PUNCT
ejde-942	205	21	1	1	NUM
ejde-942	205	22	α1	α1	PROPN
ejde-942	205	23	√	√	INTJ
ejde-942	205	24	λk	λk	ADP
ejde-942	205	25	−	−	PROPN
ejde-942	205	26	ρ0	ρ0	PROPN
ejde-942	205	27	)	)	PUNCT
ejde-942	205	28	∈	∈	PROPN
ejde-942	205	29	(	(	PUNCT
ejde-942	205	30	0	0	NUM
ejde-942	205	31	,	,	PUNCT
ejde-942	205	32	π	π	PROPN
ejde-942	205	33	2	2	NUM
ejde-942	205	34	)	)	PUNCT
ejde-942	205	35	,	,	PUNCT
ejde-942	205	36	i.e.	i.e.	X
ejde-942	205	37	,	,	PUNCT
ejde-942	205	38	θ(π)−	θ(π)−	VERB
ejde-942	205	39	θ(0	θ(0	PROPN
ejde-942	205	40	)	)	PUNCT
ejde-942	205	41	>	>	X
ejde-942	206	1	(	(	PUNCT
ejde-942	206	2	k	k	X
ejde-942	206	3	−	−	PROPN
ejde-942	206	4	1	1	NUM
ejde-942	206	5	2	2	NUM
ejde-942	206	6	)	)	PUNCT
ejde-942	206	7	π	π	PROPN
ejde-942	206	8	.	.	PUNCT
ejde-942	207	1	thus	thus	ADV
ejde-942	207	2	,	,	PUNCT
ejde-942	207	3	we	we	PRON
ejde-942	207	4	conclude	conclude	VERB
ejde-942	207	5	that	that	SCONJ
ejde-942	207	6	λk	λk	ADP
ejde-942	207	7	≥	≥	PRON
ejde-942	207	8	(	(	PUNCT
ejde-942	207	9	k	k	NOUN
ejde-942	207	10	−	−	PROPN
ejde-942	207	11	1	1	NUM
ejde-942	207	12	2	2	NUM
ejde-942	207	13	)	)	PUNCT
ejde-942	207	14	2	2	NUM
ejde-942	207	15	+	+	CCONJ
ejde-942	207	16	ρ0	ρ0	PROPN
ejde-942	207	17	secondly	secondly	ADV
ejde-942	207	18	,	,	PUNCT
ejde-942	207	19	we	we	PRON
ejde-942	207	20	prove	prove	VERB
ejde-942	207	21	the	the	DET
ejde-942	207	22	upper	upper	ADJ
ejde-942	207	23	bound	bound	NOUN
ejde-942	207	24	of	of	ADP
ejde-942	207	25	λk	λk	PROPN
ejde-942	207	26	.	.	PUNCT
ejde-942	208	1	let	let	VERB
ejde-942	208	2	pk	pk	NOUN
ejde-942	208	3	=	=	VERB
ejde-942	208	4	√	√	PROPN
ejde-942	208	5	λk	λk	ADP
ejde-942	208	6	−	−	PROPN
ejde-942	208	7	ρ0	ρ0	PROPN
ejde-942	208	8	.	.	PUNCT
ejde-942	209	1	then	then	ADV
ejde-942	209	2	,	,	PUNCT
ejde-942	209	3	as	as	ADP
ejde-942	209	4	an	an	DET
ejde-942	209	5	increasing	increase	VERB
ejde-942	209	6	function	function	NOUN
ejde-942	209	7	of	of	ADP
ejde-942	209	8	k	k	PROPN
ejde-942	209	9	,	,	PUNCT
ejde-942	209	10	pk	pk	NOUN
ejde-942	209	11	approaches	approach	NOUN
ejde-942	209	12	infinity	infinity	NOUN
ejde-942	209	13	and	and	CCONJ
ejde-942	209	14	satisfies	satisfie	NOUN
ejde-942	209	15	pk	pk	PROPN
ejde-942	209	16	>	>	X
ejde-942	209	17	1/2	1/2	NUM
ejde-942	209	18	.	.	PUNCT
ejde-942	210	1	notice	notice	VERB
ejde-942	210	2	that	that	SCONJ
ejde-942	210	3	lim	lim	PROPN
ejde-942	210	4	k→+∞	k→+∞	PROPN
ejde-942	210	5	pk	pk	PROPN
ejde-942	210	6	(	(	PUNCT
ejde-942	210	7	arctan	arctan	PROPN
ejde-942	210	8	(	(	PUNCT
ejde-942	210	9	pk	pk	NOUN
ejde-942	210	10	α1	α1	PROPN
ejde-942	210	11	)	)	PUNCT
ejde-942	210	12	−	−	PROPN
ejde-942	211	1	π	π	NOUN
ejde-942	211	2	2	2	X
ejde-942	211	3	)	)	PUNCT
ejde-942	211	4	=	=	PUNCT
ejde-942	211	5	−α1	−α1	PROPN
ejde-942	211	6	.	.	PUNCT
ejde-942	211	7	define	define	VERB
ejde-942	211	8	f(p	f(p	PROPN
ejde-942	211	9	)	)	PUNCT
ejde-942	211	10	:	:	PUNCT
ejde-942	212	1	=	=	SYM
ejde-942	212	2	p	p	X
ejde-942	212	3	(	(	PUNCT
ejde-942	212	4	arctan	arctan	PROPN
ejde-942	212	5	(	(	PUNCT
ejde-942	212	6	p	p	PROPN
ejde-942	212	7	α1	α1	PROPN
ejde-942	212	8	)	)	PUNCT
ejde-942	212	9	−	−	PROPN
ejde-942	213	1	π	π	PROPN
ejde-942	213	2	2	2	NUM
ejde-942	213	3	)	)	PUNCT
ejde-942	213	4	.	.	PUNCT
ejde-942	214	1	taking	take	VERB
ejde-942	214	2	the	the	DET
ejde-942	214	3	derivative	derivative	NOUN
ejde-942	214	4	of	of	ADP
ejde-942	214	5	f(p	f(p	PROPN
ejde-942	214	6	)	)	PUNCT
ejde-942	214	7	,	,	PUNCT
ejde-942	214	8	we	we	PRON
ejde-942	214	9	obtain	obtain	VERB
ejde-942	214	10	f	f	PROPN
ejde-942	214	11	′(p	′(p	NOUN
ejde-942	214	12	)	)	PUNCT
ejde-942	214	13	=	=	SYM
ejde-942	214	14	arctan	arctan	PROPN
ejde-942	214	15	(	(	PUNCT
ejde-942	214	16	p	p	PROPN
ejde-942	214	17	α1	α1	PROPN
ejde-942	214	18	)	)	PUNCT
ejde-942	214	19	−	−	PROPN
ejde-942	215	1	π	π	NOUN
ejde-942	215	2	2	2	NUM
ejde-942	215	3	+	+	NUM
ejde-942	215	4	α1p	α1p	NOUN
ejde-942	215	5	p2	p2	NOUN
ejde-942	215	6	+	+	CCONJ
ejde-942	215	7	α2	α2	ADJ
ejde-942	215	8	1	1	NUM
ejde-942	215	9	.	.	PUNCT
ejde-942	216	1	and	and	CCONJ
ejde-942	216	2	limp→+∞	limp→+∞	ADP
ejde-942	216	3	f	f	PROPN
ejde-942	216	4	′(p	′(p	PROPN
ejde-942	216	5	)	)	PUNCT
ejde-942	216	6	=	=	SYM
ejde-942	217	1	0	0	X
ejde-942	217	2	.	.	PUNCT
ejde-942	218	1	since	since	SCONJ
ejde-942	218	2	f	f	PROPN
ejde-942	218	3	′′(p	′′(p	PROPN
ejde-942	218	4	)	)	PUNCT
ejde-942	218	5	=	=	SYM
ejde-942	219	1	2α1	2α1	NUM
ejde-942	219	2	p2	p2	NOUN
ejde-942	219	3	+	+	CCONJ
ejde-942	219	4	α2	α2	ADJ
ejde-942	219	5	1	1	NUM
ejde-942	219	6	−	−	PROPN
ejde-942	219	7	2α1p	2α1p	NOUN
ejde-942	219	8	2	2	NUM
ejde-942	219	9	(	(	PUNCT
ejde-942	219	10	p2	p2	X
ejde-942	219	11	+	+	CCONJ
ejde-942	219	12	α2	α2	ADJ
ejde-942	219	13	1	1	NUM
ejde-942	219	14	)	)	PUNCT
ejde-942	219	15	2	2	NUM
ejde-942	219	16	=	=	SYM
ejde-942	219	17	2α3	2α3	NUM
ejde-942	219	18	1	1	NUM
ejde-942	219	19	(	(	PUNCT
ejde-942	219	20	p2	p2	X
ejde-942	219	21	+	+	CCONJ
ejde-942	219	22	α2	α2	ADJ
ejde-942	219	23	1	1	NUM
ejde-942	219	24	)	)	PUNCT
ejde-942	219	25	2	2	NUM
ejde-942	219	26	>	>	SYM
ejde-942	219	27	0	0	NUM
ejde-942	219	28	,	,	PUNCT
ejde-942	219	29	ejde-2024/53	ejde-2024/53	ADP
ejde-942	219	30	deep	deep	ADJ
ejde-942	219	31	learning	learning	NOUN
ejde-942	219	32	method	method	NOUN
ejde-942	219	33	for	for	ADP
ejde-942	219	34	finding	find	VERB
ejde-942	219	35	eigenpairs	eigenpair	NOUN
ejde-942	219	36	9	9	NUM
ejde-942	219	37	we	we	PRON
ejde-942	219	38	obtain	obtain	VERB
ejde-942	219	39	f	f	PROPN
ejde-942	219	40	′(p	′(p	NOUN
ejde-942	219	41	)	)	PUNCT
ejde-942	219	42	<	<	X
ejde-942	219	43	0	0	NUM
ejde-942	219	44	,	,	PUNCT
ejde-942	219	45	p	p	PROPN
ejde-942	219	46	∈	∈	PROPN
ejde-942	219	47	[	[	PUNCT
ejde-942	219	48	12	12	NUM
ejde-942	219	49	,	,	PUNCT
ejde-942	219	50	+	+	NOUN
ejde-942	219	51	∞	∞	NOUN
ejde-942	219	52	)	)	PUNCT
ejde-942	219	53	.	.	PUNCT
ejde-942	220	1	it	it	PRON
ejde-942	220	2	implies	imply	VERB
ejde-942	220	3	that√	that√	NOUN
ejde-942	220	4	λk	λk	ADP
ejde-942	220	5	−	−	PROPN
ejde-942	220	6	ρ0	ρ0	PROPN
ejde-942	220	7	(	(	PUNCT
ejde-942	220	8	arctan	arctan	PROPN
ejde-942	220	9	(	(	PUNCT
ejde-942	220	10	√λk	√λk	NOUN
ejde-942	220	11	−	−	PROPN
ejde-942	220	12	ρ0	ρ0	PROPN
ejde-942	220	13	α1	α1	PROPN
ejde-942	220	14	)	)	PUNCT
ejde-942	220	15	−	−	PROPN
ejde-942	221	1	π	π	PROPN
ejde-942	221	2	2	2	NUM
ejde-942	221	3	)	)	PUNCT
ejde-942	221	4	≥	≥	NOUN
ejde-942	221	5	−α1	−α1	NOUN
ejde-942	221	6	.	.	PROPN
ejde-942	222	1	hence	hence	ADV
ejde-942	222	2	θ(0	θ(0	PROPN
ejde-942	222	3	)	)	PUNCT
ejde-942	222	4	≥	≥	NOUN
ejde-942	222	5	π	π	NOUN
ejde-942	222	6	2	2	NUM
ejde-942	222	7	−	−	PROPN
ejde-942	222	8	α1√	α1√	PROPN
ejde-942	222	9	λk	λk	ADP
ejde-942	222	10	−	−	NOUN
ejde-942	222	11	ρ0	ρ0	PROPN
ejde-942	222	12	.	.	PUNCT
ejde-942	223	1	by	by	ADP
ejde-942	223	2	the	the	DET
ejde-942	223	3	prüfer	prüfer	NOUN
ejde-942	223	4	transformation	transformation	NOUN
ejde-942	223	5	,	,	PUNCT
ejde-942	223	6	we	we	PRON
ejde-942	223	7	obtain	obtain	VERB
ejde-942	223	8	θ′	θ′	NOUN
ejde-942	223	9	=	=	SYM
ejde-942	223	10	√	√	NUM
ejde-942	224	1	λk	λk	ADP
ejde-942	224	2	−	−	PROPN
ejde-942	224	3	ρ0	ρ0	PROPN
ejde-942	224	4	cos	cos	PROPN
ejde-942	224	5	2	2	NUM
ejde-942	224	6	θ	θ	NOUN
ejde-942	224	7	−	−	PROPN
ejde-942	224	8	(	(	PUNCT
ejde-942	224	9	ηu(x)−	ηu(x)−	PROPN
ejde-942	224	10	ρ0	ρ0	PROPN
ejde-942	224	11	)	)	PUNCT
ejde-942	224	12	sin	sin	NOUN
ejde-942	224	13	2	2	NUM
ejde-942	224	14	θ√	θ√	PROPN
ejde-942	224	15	λk	λk	ADP
ejde-942	224	16	−	−	PROPN
ejde-942	225	1	ρ0	ρ0	PROPN
ejde-942	225	2	≥	≥	NOUN
ejde-942	225	3	√	√	NOUN
ejde-942	225	4	λk	λk	ADP
ejde-942	225	5	−	−	PROPN
ejde-942	225	6	ρ0	ρ0	PROPN
ejde-942	225	7	−	−	PROPN
ejde-942	225	8	{	{	PUNCT
ejde-942	225	9	ηu(x)−	ηu(x)−	PROPN
ejde-942	225	10	ρ0}+	ρ0}+	PROPN
ejde-942	225	11	sin2	sin2	PROPN
ejde-942	225	12	θ√	θ√	PROPN
ejde-942	226	1	λk	λk	ADV
ejde-942	226	2	−	−	PROPN
ejde-942	226	3	ρ0	ρ0	PROPN
ejde-942	226	4	≥	≥	NOUN
ejde-942	227	1	√	√	NOUN
ejde-942	227	2	λk	λk	ADP
ejde-942	227	3	−	−	PROPN
ejde-942	227	4	ρ0	ρ0	PROPN
ejde-942	227	5	−	−	PROPN
ejde-942	227	6	{	{	PUNCT
ejde-942	227	7	ηu(x)−	ηu(x)−	PROPN
ejde-942	227	8	ρ0}+√	ρ0}+√	PROPN
ejde-942	228	1	λk	λk	ADP
ejde-942	228	2	−	−	PROPN
ejde-942	228	3	ρ0	ρ0	PROPN
ejde-942	228	4	.	.	PUNCT
ejde-942	229	1	by	by	ADP
ejde-942	229	2	integrating	integrate	VERB
ejde-942	229	3	both	both	DET
ejde-942	229	4	sides	side	NOUN
ejde-942	229	5	over	over	ADP
ejde-942	229	6	the	the	DET
ejde-942	229	7	interval	interval	NOUN
ejde-942	229	8	(	(	PUNCT
ejde-942	229	9	0	0	NUM
ejde-942	229	10	,	,	PUNCT
ejde-942	229	11	π	π	PROPN
ejde-942	229	12	)	)	PUNCT
ejde-942	229	13	,	,	PUNCT
ejde-942	229	14	we	we	PRON
ejde-942	229	15	obtain	obtain	VERB
ejde-942	229	16	:	:	PUNCT
ejde-942	229	17	θ(π)−	θ(π)−	PROPN
ejde-942	229	18	θ(0	θ(0	PROPN
ejde-942	229	19	)	)	PUNCT
ejde-942	229	20	≥	≥	NOUN
ejde-942	230	1	√	√	NOUN
ejde-942	230	2	λk	λk	ADP
ejde-942	230	3	−	−	PROPN
ejde-942	230	4	ρ0π	ρ0π	ADJ
ejde-942	230	5	−	−	PROPN
ejde-942	230	6	∫	∫	PROPN
ejde-942	230	7	π	π	X
ejde-942	230	8	0	0	PUNCT
ejde-942	230	9	{	{	PUNCT
ejde-942	230	10	ηu	ηu	NOUN
ejde-942	230	11	−	−	PROPN
ejde-942	230	12	ρ0}+dx√	ρ0}+dx√	PROPN
ejde-942	230	13	λk	λk	ADP
ejde-942	230	14	−	−	PROPN
ejde-942	230	15	ρ0	ρ0	PROPN
ejde-942	230	16	.	.	PUNCT
ejde-942	231	1	therefore	therefore	ADV
ejde-942	231	2	,	,	PUNCT
ejde-942	231	3	kπ	kπ	PROPN
ejde-942	231	4	−	−	PROPN
ejde-942	231	5	π	π	PROPN
ejde-942	231	6	2	2	NUM
ejde-942	231	7	+	+	X
ejde-942	231	8	α1√	α1√	NOUN
ejde-942	231	9	λk	λk	ADP
ejde-942	231	10	−	−	PROPN
ejde-942	231	11	ρ0	ρ0	PROPN
ejde-942	231	12	≥	≥	NOUN
ejde-942	231	13	√	√	NOUN
ejde-942	231	14	λk	λk	ADP
ejde-942	231	15	−	−	PROPN
ejde-942	231	16	ρ0π	ρ0π	ADJ
ejde-942	231	17	−	−	PROPN
ejde-942	231	18	∫	∫	PROPN
ejde-942	231	19	π	π	X
ejde-942	231	20	0	0	PUNCT
ejde-942	231	21	{	{	PUNCT
ejde-942	231	22	ηu	ηu	NOUN
ejde-942	231	23	−	−	PROPN
ejde-942	231	24	ρ0}+dx√	ρ0}+dx√	PROPN
ejde-942	231	25	λk	λk	ADP
ejde-942	231	26	−	−	PROPN
ejde-942	231	27	ρ0	ρ0	PROPN
ejde-942	231	28	.	.	PUNCT
ejde-942	232	1	defining	define	VERB
ejde-942	232	2	d	d	NOUN
ejde-942	232	3	:	:	PUNCT
ejde-942	232	4	=	=	X
ejde-942	232	5	λk	λk	ADP
ejde-942	232	6	−	−	PROPN
ejde-942	232	7	ρ0	ρ0	PROPN
ejde-942	232	8	,	,	PUNCT
ejde-942	232	9	b	b	NOUN
ejde-942	232	10	:	:	PUNCT
ejde-942	233	1	=	=	SYM
ejde-942	233	2	k	k	NOUN
ejde-942	234	1	−	−	NUM
ejde-942	235	1	1	1	NUM
ejde-942	235	2	2	2	NUM
ejde-942	235	3	,	,	PUNCT
ejde-942	235	4	c	c	NOUN
ejde-942	235	5	:	:	PUNCT
ejde-942	235	6	=	=	SYM
ejde-942	235	7	1	1	NUM
ejde-942	235	8	π	π	X
ejde-942	235	9	(	(	PUNCT
ejde-942	235	10	∫	∫	PROPN
ejde-942	235	11	π	π	PROPN
ejde-942	235	12	0	0	NUM
ejde-942	235	13	{	{	PUNCT
ejde-942	235	14	ηu(x	ηu(x	NOUN
ejde-942	235	15	)	)	PUNCT
ejde-942	235	16	−	−	ADP
ejde-942	235	17	ρ0}+dx	ρ0}+dx	NUM
ejde-942	235	18	+	+	NUM
ejde-942	235	19	α1	α1	NOUN
ejde-942	235	20	)	)	PUNCT
ejde-942	235	21	,	,	PUNCT
ejde-942	235	22	we	we	PRON
ejde-942	235	23	obtain	obtain	VERB
ejde-942	235	24	d	d	NOUN
ejde-942	235	25	≤	≤	NUM
ejde-942	235	26	b	b	NOUN
ejde-942	235	27	√	√	PROPN
ejde-942	235	28	d	d	NOUN
ejde-942	236	1	+	+	CCONJ
ejde-942	236	2	c	c	X
ejde-942	236	3	,	,	PUNCT
ejde-942	236	4	which	which	PRON
ejde-942	236	5	implies	imply	VERB
ejde-942	236	6	√	√	PROPN
ejde-942	236	7	d	d	SYM
ejde-942	236	8	≤	≤	X
ejde-942	236	9	(	(	PUNCT
ejde-942	236	10	√	√	ADP
ejde-942	236	11	b2	b2	NOUN
ejde-942	236	12	+	+	CCONJ
ejde-942	236	13	4c	4c	NOUN
ejde-942	237	1	+	+	NOUN
ejde-942	238	1	b)/2	b)/2	PROPN
ejde-942	238	2	.	.	PUNCT
ejde-942	239	1	by	by	ADP
ejde-942	239	2	inequalities	inequality	NOUN
ejde-942	239	3	√	√	PROPN
ejde-942	239	4	b2	b2	PROPN
ejde-942	239	5	+	+	CCONJ
ejde-942	239	6	4c	4c	NOUN
ejde-942	239	7	=	=	SYM
ejde-942	239	8	b	b	NOUN
ejde-942	239	9	√	√	ADJ
ejde-942	239	10	1	1	NUM
ejde-942	239	11	+	+	NUM
ejde-942	239	12	4c	4c	NUM
ejde-942	239	13	b2	b2	NOUN
ejde-942	239	14	≤	≤	NOUN
ejde-942	239	15	b(1	b(1	PROPN
ejde-942	239	16	+	+	CCONJ
ejde-942	239	17	2c	2c	NUM
ejde-942	239	18	b2	b2	NOUN
ejde-942	239	19	)	)	PUNCT
ejde-942	240	1	=	=	SYM
ejde-942	240	2	b	b	X
ejde-942	240	3	+	+	NUM
ejde-942	240	4	2c	2c	NUM
ejde-942	240	5	b	b	NOUN
ejde-942	240	6	,	,	PUNCT
ejde-942	240	7	we	we	PRON
ejde-942	240	8	obtain	obtain	VERB
ejde-942	240	9	the	the	DET
ejde-942	240	10	upper	upper	ADJ
ejde-942	240	11	bound	bind	VERB
ejde-942	240	12	d	d	PROPN
ejde-942	240	13	≤	≤	NOUN
ejde-942	240	14	b2	b2	NOUN
ejde-942	240	15	+	+	CCONJ
ejde-942	240	16	2b	2b	NUM
ejde-942	240	17	√	√	ADP
ejde-942	240	18	b2	b2	NOUN
ejde-942	240	19	+	+	CCONJ
ejde-942	240	20	4c	4c	NOUN
ejde-942	240	21	+	+	NOUN
ejde-942	240	22	b2	b2	NOUN
ejde-942	240	23	+	+	CCONJ
ejde-942	240	24	4c	4c	NUM
ejde-942	240	25	4	4	NUM
ejde-942	240	26	≤	≤	NOUN
ejde-942	240	27	b2	b2	NOUN
ejde-942	240	28	+	+	CCONJ
ejde-942	240	29	2c	2c	NOUN
ejde-942	240	30	.	.	PUNCT
ejde-942	241	1	thus	thus	ADV
ejde-942	241	2	,	,	PUNCT
ejde-942	241	3	we	we	PRON
ejde-942	241	4	conclude	conclude	VERB
ejde-942	241	5	that	that	SCONJ
ejde-942	241	6	λk	λk	ADP
ejde-942	241	7	≤	≤	PROPN
ejde-942	241	8	(	(	PUNCT
ejde-942	241	9	k	k	NOUN
ejde-942	241	10	−	−	PROPN
ejde-942	241	11	1	1	NUM
ejde-942	241	12	2	2	NUM
ejde-942	241	13	)	)	SYM
ejde-942	241	14	2	2	NUM
ejde-942	241	15	+	+	NUM
ejde-942	241	16	ρ1	ρ1	NOUN
ejde-942	241	17	+	+	CCONJ
ejde-942	241	18	2α1	2α1	NUM
ejde-942	241	19	π	π	NOUN
ejde-942	241	20	.	.	PUNCT
ejde-942	242	1	the	the	DET
ejde-942	242	2	aforementioned	aforementioned	ADJ
ejde-942	242	3	conclusion	conclusion	NOUN
ejde-942	242	4	can	can	AUX
ejde-942	242	5	be	be	AUX
ejde-942	242	6	easily	easily	ADV
ejde-942	242	7	extend	extend	VERB
ejde-942	242	8	to	to	ADP
ejde-942	242	9	the	the	DET
ejde-942	242	10	case	case	NOUN
ejde-942	242	11	where	where	SCONJ
ejde-942	242	12	a1	a1	NOUN
ejde-942	242	13	=	=	NOUN
ejde-942	242	14	0	0	PROPN
ejde-942	242	15	and	and	CCONJ
ejde-942	242	16	α2	α2	ADJ
ejde-942	242	17	≥	≥	NOUN
ejde-942	242	18	0	0	NUM
ejde-942	242	19	.	.	PUNCT
ejde-942	243	1	we	we	PRON
ejde-942	243	2	omit	omit	VERB
ejde-942	243	3	the	the	DET
ejde-942	243	4	proof	proof	NOUN
ejde-942	243	5	for	for	ADP
ejde-942	243	6	brevity	brevity	NOUN
ejde-942	243	7	.	.	PUNCT
ejde-942	244	1	□	□	PUNCT
ejde-942	244	2	lemma	lemma	PROPN
ejde-942	244	3	3.3	3.3	NUM
ejde-942	244	4	.	.	PUNCT
ejde-942	245	1	assume	assume	VERB
ejde-942	245	2	that	that	SCONJ
ejde-942	245	3	α1	α1	PROPN
ejde-942	245	4	≥	≥	NOUN
ejde-942	245	5	0	0	NUM
ejde-942	245	6	and	and	CCONJ
ejde-942	245	7	α2	α2	ADJ
ejde-942	245	8	≥	≥	NOUN
ejde-942	245	9	0	0	NUM
ejde-942	245	10	.	.	PUNCT
ejde-942	246	1	then	then	ADV
ejde-942	246	2	the	the	DET
ejde-942	246	3	sturm	sturm	PROPN
ejde-942	246	4	-	-	PUNCT
ejde-942	246	5	liouville	liouville	NOUN
ejde-942	246	6	problems	problem	NOUN
ejde-942	246	7	(	(	PUNCT
ejde-942	246	8	3.1	3.1	NUM
ejde-942	246	9	)	)	PUNCT
ejde-942	246	10	with	with	ADP
ejde-942	246	11	case	case	NOUN
ejde-942	246	12	(	(	PUNCT
ejde-942	246	13	2	2	NUM
ejde-942	246	14	)	)	PUNCT
ejde-942	246	15	,	,	PUNCT
ejde-942	246	16	(	(	PUNCT
ejde-942	246	17	5	5	NUM
ejde-942	246	18	)	)	PUNCT
ejde-942	246	19	and	and	CCONJ
ejde-942	246	20	(	(	PUNCT
ejde-942	246	21	6	6	X
ejde-942	246	22	)	)	PUNCT
ejde-942	246	23	satisfy	satisfy	NOUN
ejde-942	246	24	(	(	PUNCT
ejde-942	246	25	k	k	PROPN
ejde-942	246	26	−	−	PROPN
ejde-942	247	1	1)2	1)2	NUM
ejde-942	247	2	+	+	CCONJ
ejde-942	247	3	ρ0	ρ0	PROPN
ejde-942	247	4	≤	≤	PROPN
ejde-942	247	5	λk	λk	ADP
ejde-942	247	6	≤	≤	PROPN
ejde-942	247	7	(	(	PUNCT
ejde-942	247	8	k	k	PROPN
ejde-942	248	1	−	−	PROPN
ejde-942	248	2	1)2	1)2	NUM
ejde-942	248	3	+	+	NUM
ejde-942	248	4	ρ1	ρ1	NOUN
ejde-942	248	5	+	+	CCONJ
ejde-942	248	6	2α1	2α1	NUM
ejde-942	248	7	π	π	X
ejde-942	248	8	+	+	CCONJ
ejde-942	248	9	2α2	2α2	NUM
ejde-942	248	10	π	π	NOUN
ejde-942	248	11	,	,	PUNCT
ejde-942	248	12	k	k	PROPN
ejde-942	248	13	∈	∈	PROPN
ejde-942	248	14	n+	n+	PROPN
ejde-942	248	15	.	.	PUNCT
ejde-942	249	1	the	the	DET
ejde-942	249	2	above	above	ADJ
ejde-942	249	3	lemma	lemma	PROPN
ejde-942	249	4	can	can	AUX
ejde-942	249	5	be	be	AUX
ejde-942	249	6	proved	prove	VERB
ejde-942	249	7	by	by	ADP
ejde-942	249	8	a	a	DET
ejde-942	249	9	similar	similar	ADJ
ejde-942	249	10	argument	argument	NOUN
ejde-942	249	11	to	to	ADP
ejde-942	249	12	that	that	PRON
ejde-942	249	13	of	of	ADP
ejde-942	249	14	lemma	lemma	PROPN
ejde-942	249	15	3.2	3.2	NUM
ejde-942	249	16	.	.	PUNCT
ejde-942	250	1	now	now	ADV
ejde-942	250	2	,	,	PUNCT
ejde-942	250	3	we	we	PRON
ejde-942	250	4	give	give	VERB
ejde-942	250	5	a	a	DET
ejde-942	250	6	rough	rough	ADJ
ejde-942	250	7	estimation	estimation	NOUN
ejde-942	250	8	of	of	ADP
ejde-942	250	9	λk	λk	PRON
ejde-942	250	10	obtained	obtain	VERB
ejde-942	250	11	by	by	ADP
ejde-942	250	12	the	the	DET
ejde-942	250	13	neural	neural	ADJ
ejde-942	250	14	network	network	NOUN
ejde-942	250	15	method	method	NOUN
ejde-942	250	16	for	for	ADP
ejde-942	250	17	the	the	DET
ejde-942	250	18	sturm	sturm	PROPN
ejde-942	250	19	-	-	PUNCT
ejde-942	250	20	liouville	liouville	NOUN
ejde-942	250	21	eigenvalue	eigenvalue	NOUN
ejde-942	250	22	problems	problem	NOUN
ejde-942	250	23	.	.	PUNCT
ejde-942	251	1	10	10	NUM
ejde-942	251	2	s.	s.	PROPN
ejde-942	251	3	zhang	zhang	PROPN
ejde-942	251	4	,	,	PUNCT
ejde-942	251	5	j.	j.	PROPN
ejde-942	251	6	zu	zu	PROPN
ejde-942	251	7	,	,	PUNCT
ejde-942	251	8	j.	j.	PROPN
ejde-942	251	9	zhang	zhang	PROPN
ejde-942	251	10	ejde-2024/53	ejde-2024/53	PROPN
ejde-942	251	11	proposition	proposition	NOUN
ejde-942	251	12	3.4	3.4	NUM
ejde-942	251	13	.	.	PUNCT
ejde-942	251	14	assume	assume	VERB
ejde-942	251	15	that	that	SCONJ
ejde-942	251	16	ρ1−	ρ1−	PROPN
ejde-942	251	17	ρ0	ρ0	PROPN
ejde-942	251	18	≤	≤	ADJ
ejde-942	251	19	1	1	NUM
ejde-942	251	20	.	.	PUNCT
ejde-942	252	1	if	if	SCONJ
ejde-942	252	2	λk	λk	PROPN
ejde-942	252	3	∈	∈	PROPN
ejde-942	253	1	[	[	X
ejde-942	253	2	k2	k2	PROPN
ejde-942	253	3	+	+	CCONJ
ejde-942	253	4	ρ0	ρ0	PROPN
ejde-942	253	5	,	,	PUNCT
ejde-942	253	6	k	k	PROPN
ejde-942	253	7	2	2	NUM
ejde-942	253	8	+	+	NUM
ejde-942	253	9	ρ1	ρ1	NOUN
ejde-942	253	10	]	]	PUNCT
ejde-942	253	11	,	,	PUNCT
ejde-942	253	12	k	k	PROPN
ejde-942	253	13	∈	∈	PROPN
ejde-942	253	14	n+	n+	PROPN
ejde-942	253	15	,	,	PUNCT
ejde-942	253	16	the	the	DET
ejde-942	253	17	sturm	sturm	NOUN
ejde-942	253	18	-	-	PUNCT
ejde-942	253	19	liouville	liouville	NOUN
ejde-942	253	20	eigenvalue	eigenvalue	NOUN
ejde-942	253	21	problems	problem	NOUN
ejde-942	253	22	with	with	ADP
ejde-942	253	23	case	case	NOUN
ejde-942	253	24	(	(	PUNCT
ejde-942	253	25	1	1	X
ejde-942	253	26	)	)	PUNCT
ejde-942	253	27	imply	imply	VERB
ejde-942	253	28	that	that	SCONJ
ejde-942	253	29	|λk	|λk	NUM
ejde-942	253	30	−	−	PROPN
ejde-942	253	31	λk|	λk|	NOUN
ejde-942	253	32	≤	≤	NUM
ejde-942	253	33	1	1	NUM
ejde-942	253	34	≤	≤	NUM
ejde-942	253	35	min	min	NOUN
ejde-942	253	36	j	j	PROPN
ejde-942	253	37	̸=k	̸=k	PROPN
ejde-942	253	38	{	{	PUNCT
ejde-942	253	39	|λk	|λk	PROPN
ejde-942	253	40	−	−	NOUN
ejde-942	253	41	λj	λj	PROPN
ejde-942	253	42	|	|	NOUN
ejde-942	253	43	}	}	PUNCT
ejde-942	253	44	.	.	PUNCT
ejde-942	254	1	proof	proof	NOUN
ejde-942	254	2	.	.	PUNCT
ejde-942	255	1	by	by	ADP
ejde-942	255	2	lemma	lemma	PROPN
ejde-942	255	3	3.1	3.1	NUM
ejde-942	255	4	,	,	PUNCT
ejde-942	255	5	we	we	PRON
ejde-942	255	6	have	have	VERB
ejde-942	255	7	|λk	|λk	NUM
ejde-942	255	8	−	−	PROPN
ejde-942	255	9	λk|	λk|	PROPN
ejde-942	255	10	≤	≤	PROPN
ejde-942	255	11	|ρ1	|ρ1	NOUN
ejde-942	255	12	−	−	PROPN
ejde-942	255	13	ρ0|	ρ0|	PROPN
ejde-942	255	14	≤	≤	NUM
ejde-942	255	15	1	1	NUM
ejde-942	255	16	.	.	PUNCT
ejde-942	256	1	for	for	ADP
ejde-942	256	2	j	j	PROPN
ejde-942	256	3	>	>	X
ejde-942	256	4	k	k	PROPN
ejde-942	256	5	≥	≥	NUM
ejde-942	256	6	1	1	NUM
ejde-942	256	7	,	,	PUNCT
ejde-942	256	8	we	we	PRON
ejde-942	256	9	have	have	VERB
ejde-942	256	10	|λk	|λk	NUM
ejde-942	256	11	−	−	NOUN
ejde-942	257	1	λj	λj	INTJ
ejde-942	257	2	|	|	ADV
ejde-942	257	3	≥	≥	AUX
ejde-942	257	4	|λj	|λj	PUNCT
ejde-942	258	1	−	−	PROPN
ejde-942	259	1	λk|	λk|	X
ejde-942	259	2	−	−	PROPN
ejde-942	259	3	|λk	|λk	NUM
ejde-942	260	1	−	−	PROPN
ejde-942	260	2	λk|	λk|	X
ejde-942	260	3	≥	≥	X
ejde-942	260	4	|j2	|j2	NOUN
ejde-942	261	1	+	+	CCONJ
ejde-942	261	2	ρ0	ρ0	PROPN
ejde-942	261	3	−	−	PROPN
ejde-942	261	4	k2	k2	PROPN
ejde-942	261	5	−	−	PROPN
ejde-942	261	6	ρ1|	ρ1|	PROPN
ejde-942	261	7	−	−	PROPN
ejde-942	262	1	|k2	|k2	NOUN
ejde-942	262	2	+	+	CCONJ
ejde-942	262	3	ρ1	ρ1	PROPN
ejde-942	262	4	−	−	PROPN
ejde-942	262	5	k2	k2	PROPN
ejde-942	262	6	−	−	PROPN
ejde-942	262	7	ρ0|	ρ0|	NUM
ejde-942	262	8	≥	≥	NOUN
ejde-942	262	9	|j2	|j2	NOUN
ejde-942	262	10	−	−	PUNCT
ejde-942	262	11	k2|	k2|	X
ejde-942	262	12	−	−	NOUN
ejde-942	262	13	2|ρ1	2|ρ1	NUM
ejde-942	262	14	−	−	PROPN
ejde-942	262	15	ρ0|	ρ0|	NUM
ejde-942	262	16	≥	≥	NUM
ejde-942	262	17	1	1	NUM
ejde-942	262	18	.	.	PUNCT
ejde-942	263	1	the	the	DET
ejde-942	263	2	second	second	ADJ
ejde-942	263	3	inequality	inequality	NOUN
ejde-942	263	4	follows	follow	VERB
ejde-942	263	5	from	from	ADP
ejde-942	263	6	lemma	lemma	PROPN
ejde-942	263	7	3.1	3.1	NUM
ejde-942	263	8	,	,	PUNCT
ejde-942	263	9	and	and	CCONJ
ejde-942	263	10	the	the	DET
ejde-942	263	11	fourth	fourth	ADJ
ejde-942	263	12	inequality	inequality	NOUN
ejde-942	263	13	follows	follow	VERB
ejde-942	263	14	from	from	ADP
ejde-942	263	15	the	the	DET
ejde-942	263	16	fact	fact	NOUN
ejde-942	263	17	that	that	SCONJ
ejde-942	263	18	j2	j2	PROPN
ejde-942	263	19	−	−	PROPN
ejde-942	263	20	k2	k2	PROPN
ejde-942	263	21	≥	≥	PROPN
ejde-942	263	22	j	j	PROPN
ejde-942	263	23	+	+	CCONJ
ejde-942	263	24	k	k	PROPN
ejde-942	263	25	≥	≥	NUM
ejde-942	263	26	3	3	NUM
ejde-942	263	27	.	.	PUNCT
ejde-942	263	28	similarly	similarly	ADV
ejde-942	263	29	,	,	PUNCT
ejde-942	263	30	for	for	ADP
ejde-942	263	31	k	k	PROPN
ejde-942	263	32	>	>	X
ejde-942	263	33	j	j	PROPN
ejde-942	263	34	≥	≥	NUM
ejde-942	263	35	1	1	NUM
ejde-942	263	36	,	,	PUNCT
ejde-942	263	37	we	we	PRON
ejde-942	263	38	also	also	ADV
ejde-942	263	39	conclude	conclude	VERB
ejde-942	263	40	that	that	SCONJ
ejde-942	263	41	|λ−	|λ−	NOUN
ejde-942	264	1	λj	λj	INTJ
ejde-942	264	2	|	|	ADV
ejde-942	264	3	≥	≥	NOUN
ejde-942	264	4	1	1	NUM
ejde-942	264	5	.	.	PUNCT
ejde-942	265	1	thus	thus	ADV
ejde-942	265	2	,	,	PUNCT
ejde-942	265	3	for	for	ADP
ejde-942	265	4	j	j	PROPN
ejde-942	265	5	̸=	̸=	PROPN
ejde-942	265	6	k	k	PROPN
ejde-942	265	7	∈	∈	PROPN
ejde-942	265	8	n+	n+	PROPN
ejde-942	265	9	,	,	PUNCT
ejde-942	265	10	we	we	PRON
ejde-942	265	11	have	have	VERB
ejde-942	265	12	|λk	|λk	NUM
ejde-942	265	13	−	−	PROPN
ejde-942	265	14	λk|	λk|	PROPN
ejde-942	265	15	≤	≤	PROPN
ejde-942	265	16	|ρ1	|ρ1	NOUN
ejde-942	265	17	−	−	PROPN
ejde-942	266	1	ρ0|	ρ0|	SYM
ejde-942	266	2	≤	≤	NUM
ejde-942	266	3	1	1	NUM
ejde-942	266	4	≤	≤	NUM
ejde-942	266	5	min	min	NOUN
ejde-942	266	6	j	j	PROPN
ejde-942	266	7	{	{	PUNCT
ejde-942	266	8	|λk	|λk	PROPN
ejde-942	266	9	−	−	NOUN
ejde-942	266	10	λj	λj	PROPN
ejde-942	266	11	|	|	NOUN
ejde-942	266	12	}	}	PUNCT
ejde-942	266	13	.	.	PUNCT
ejde-942	267	1	□	□	PUNCT
ejde-942	267	2	proposition	proposition	NOUN
ejde-942	267	3	3.5	3.5	NUM
ejde-942	267	4	.	.	PUNCT
ejde-942	267	5	assume	assume	VERB
ejde-942	267	6	that	that	SCONJ
ejde-942	267	7	a2	a2	PROPN
ejde-942	267	8	=	=	SYM
ejde-942	267	9	0	0	NUM
ejde-942	267	10	,	,	PUNCT
ejde-942	267	11	α1	α1	X
ejde-942	267	12	≥	≥	NUM
ejde-942	267	13	0	0	NUM
ejde-942	267	14	and	and	CCONJ
ejde-942	267	15	ρ1−ρ0	ρ1−ρ0	NUM
ejde-942	267	16	+	+	NUM
ejde-942	267	17	2α1	2α1	NUM
ejde-942	267	18	π	π	NOUN
ejde-942	267	19	≤	≤	NUM
ejde-942	267	20	2	2	NUM
ejde-942	267	21	3	3	NUM
ejde-942	267	22	(	(	PUNCT
ejde-942	267	23	resp	resp	NOUN
ejde-942	267	24	.	.	PUNCT
ejde-942	268	1	a1	a1	NOUN
ejde-942	268	2	=	=	SYM
ejde-942	268	3	0	0	PROPN
ejde-942	268	4	,	,	PUNCT
ejde-942	268	5	α2	α2	ADJ
ejde-942	268	6	≥	≥	NOUN
ejde-942	268	7	0	0	NUM
ejde-942	268	8	and	and	CCONJ
ejde-942	268	9	ρ1	ρ1	NOUN
ejde-942	269	1	−	−	PROPN
ejde-942	269	2	ρ0	ρ0	PROPN
ejde-942	269	3	+	+	CCONJ
ejde-942	269	4	2α2	2α2	NUM
ejde-942	269	5	π	π	PROPN
ejde-942	269	6	≤	≤	NUM
ejde-942	269	7	2	2	NUM
ejde-942	269	8	3	3	NUM
ejde-942	269	9	)	)	PUNCT
ejde-942	269	10	.	.	PUNCT
ejde-942	270	1	if	if	SCONJ
ejde-942	270	2	λk	λk	PROPN
ejde-942	270	3	∈	∈	PROPN
ejde-942	270	4	[	[	PUNCT
ejde-942	270	5	(	(	PUNCT
ejde-942	270	6	k	k	NOUN
ejde-942	270	7	−	−	PROPN
ejde-942	270	8	1	1	NUM
ejde-942	270	9	2	2	NUM
ejde-942	270	10	)	)	PUNCT
ejde-942	270	11	2	2	NUM
ejde-942	270	12	+	+	CCONJ
ejde-942	270	13	ρ0	ρ0	PROPN
ejde-942	270	14	,	,	PUNCT
ejde-942	270	15	(	(	PUNCT
ejde-942	270	16	k	k	NOUN
ejde-942	270	17	−	−	PROPN
ejde-942	270	18	1	1	NUM
ejde-942	270	19	2	2	NUM
ejde-942	270	20	)	)	SYM
ejde-942	270	21	2	2	NUM
ejde-942	270	22	+	+	NUM
ejde-942	270	23	ρ1	ρ1	NOUN
ejde-942	270	24	+	+	CCONJ
ejde-942	270	25	2α1	2α1	NUM
ejde-942	270	26	π	π	NOUN
ejde-942	270	27	]	]	X
ejde-942	270	28	(	(	PUNCT
ejde-942	270	29	resp	resp	NOUN
ejde-942	270	30	.	.	PUNCT
ejde-942	271	1	λk	λk	ADP
ejde-942	271	2	∈	∈	PROPN
ejde-942	271	3	[	[	PUNCT
ejde-942	271	4	(	(	PUNCT
ejde-942	271	5	k	k	NOUN
ejde-942	271	6	−	−	PROPN
ejde-942	271	7	1	1	NUM
ejde-942	271	8	2	2	NUM
ejde-942	271	9	)	)	PUNCT
ejde-942	271	10	2	2	NUM
ejde-942	271	11	+	+	CCONJ
ejde-942	271	12	ρ0	ρ0	PROPN
ejde-942	271	13	,	,	PUNCT
ejde-942	271	14	(	(	PUNCT
ejde-942	271	15	k	k	NOUN
ejde-942	271	16	−	−	PROPN
ejde-942	271	17	1	1	NUM
ejde-942	271	18	2	2	NUM
ejde-942	271	19	)	)	SYM
ejde-942	271	20	2	2	NUM
ejde-942	271	21	+	+	NUM
ejde-942	271	22	ρ1	ρ1	NOUN
ejde-942	271	23	+	+	CCONJ
ejde-942	271	24	2α2	2α2	NUM
ejde-942	271	25	π	π	NOUN
ejde-942	271	26	]	]	X
ejde-942	271	27	)	)	PUNCT
ejde-942	271	28	,	,	PUNCT
ejde-942	271	29	k	k	PROPN
ejde-942	271	30	∈	∈	PROPN
ejde-942	271	31	n+	n+	PROPN
ejde-942	271	32	,	,	PUNCT
ejde-942	271	33	the	the	DET
ejde-942	271	34	sturm	sturm	NOUN
ejde-942	271	35	-	-	PUNCT
ejde-942	271	36	liouville	liouville	NOUN
ejde-942	271	37	eigenvalue	eigenvalue	NOUN
ejde-942	271	38	problems	problem	NOUN
ejde-942	271	39	with	with	ADP
ejde-942	271	40	case	case	NOUN
ejde-942	271	41	(	(	PUNCT
ejde-942	271	42	3	3	NUM
ejde-942	271	43	)	)	PUNCT
ejde-942	271	44	and	and	CCONJ
ejde-942	271	45	(	(	PUNCT
ejde-942	271	46	4	4	X
ejde-942	271	47	)	)	PUNCT
ejde-942	271	48	imply	imply	VERB
ejde-942	271	49	that	that	SCONJ
ejde-942	271	50	|λk	|λk	NUM
ejde-942	271	51	−	−	PROPN
ejde-942	271	52	λk|	λk|	NOUN
ejde-942	271	53	≤	≤	NUM
ejde-942	271	54	2	2	NUM
ejde-942	271	55	3	3	NUM
ejde-942	271	56	≤	≤	NUM
ejde-942	271	57	min	min	NOUN
ejde-942	271	58	j	j	PROPN
ejde-942	271	59	̸=k	̸=k	PROPN
ejde-942	271	60	{	{	PUNCT
ejde-942	271	61	|λk	|λk	PROPN
ejde-942	271	62	−	−	NOUN
ejde-942	271	63	λj	λj	PROPN
ejde-942	271	64	|	|	NOUN
ejde-942	271	65	}	}	PUNCT
ejde-942	271	66	.	.	PUNCT
ejde-942	272	1	by	by	ADP
ejde-942	272	2	a	a	DET
ejde-942	272	3	similar	similar	ADJ
ejde-942	272	4	argument	argument	NOUN
ejde-942	272	5	to	to	ADP
ejde-942	272	6	that	that	PRON
ejde-942	272	7	of	of	ADP
ejde-942	272	8	proposition	proposition	NOUN
ejde-942	272	9	3.4	3.4	NUM
ejde-942	272	10	,	,	PUNCT
ejde-942	272	11	we	we	PRON
ejde-942	272	12	can	can	AUX
ejde-942	272	13	prove	prove	VERB
ejde-942	272	14	proposition	proposition	NOUN
ejde-942	272	15	3.5	3.5	NUM
ejde-942	272	16	based	base	VERB
ejde-942	272	17	on	on	ADP
ejde-942	272	18	lemma	lemma	PROPN
ejde-942	272	19	3.2	3.2	NUM
ejde-942	272	20	.	.	PUNCT
ejde-942	273	1	proposition	proposition	NOUN
ejde-942	273	2	3.6	3.6	NUM
ejde-942	273	3	.	.	PUNCT
ejde-942	274	1	assume	assume	VERB
ejde-942	274	2	that	that	SCONJ
ejde-942	274	3	α1	α1	PROPN
ejde-942	274	4	≥	≥	NUM
ejde-942	274	5	0	0	NUM
ejde-942	274	6	,	,	PUNCT
ejde-942	274	7	α2	α2	ADJ
ejde-942	274	8	≥	≥	NOUN
ejde-942	274	9	0	0	NUM
ejde-942	274	10	and	and	CCONJ
ejde-942	274	11	ρ1	ρ1	NOUN
ejde-942	274	12	−	−	PROPN
ejde-942	274	13	ρ0	ρ0	PROPN
ejde-942	275	1	+	+	NOUN
ejde-942	275	2	2α1	2α1	NUM
ejde-942	275	3	π	π	X
ejde-942	275	4	+	+	CCONJ
ejde-942	275	5	2α2	2α2	NUM
ejde-942	275	6	π	π	PROPN
ejde-942	275	7	≤	≤	NUM
ejde-942	275	8	1	1	NUM
ejde-942	275	9	3	3	NUM
ejde-942	275	10	.	.	PUNCT
ejde-942	276	1	if	if	SCONJ
ejde-942	276	2	λk	λk	ADP
ejde-942	276	3	∈	∈	PROPN
ejde-942	277	1	[	[	X
ejde-942	277	2	(	(	PUNCT
ejde-942	277	3	k−1)2+ρ0	k−1)2+ρ0	PROPN
ejde-942	277	4	,	,	PUNCT
ejde-942	277	5	(	(	PUNCT
ejde-942	277	6	k−1)2+ρ1	k−1)2+ρ1	PROPN
ejde-942	277	7	+	+	PROPN
ejde-942	277	8	2α1	2α1	NUM
ejde-942	277	9	π	π	NOUN
ejde-942	277	10	+	+	CCONJ
ejde-942	277	11	2α2	2α2	NUM
ejde-942	277	12	π	π	NOUN
ejde-942	277	13	]	]	X
ejde-942	277	14	,	,	PUNCT
ejde-942	277	15	k	k	PROPN
ejde-942	277	16	∈	∈	PROPN
ejde-942	277	17	n+	n+	PROPN
ejde-942	277	18	,	,	PUNCT
ejde-942	277	19	the	the	DET
ejde-942	277	20	sturm	sturm	NOUN
ejde-942	277	21	-	-	PUNCT
ejde-942	277	22	liouville	liouville	NOUN
ejde-942	277	23	eigenvalue	eigenvalue	NOUN
ejde-942	277	24	problems	problem	NOUN
ejde-942	277	25	with	with	ADP
ejde-942	277	26	case	case	NOUN
ejde-942	277	27	(	(	PUNCT
ejde-942	277	28	2	2	NUM
ejde-942	277	29	)	)	PUNCT
ejde-942	277	30	,	,	PUNCT
ejde-942	277	31	(	(	PUNCT
ejde-942	277	32	5	5	NUM
ejde-942	277	33	)	)	PUNCT
ejde-942	277	34	and	and	CCONJ
ejde-942	277	35	(	(	PUNCT
ejde-942	277	36	6	6	X
ejde-942	277	37	)	)	PUNCT
ejde-942	277	38	imply	imply	VERB
ejde-942	277	39	that	that	SCONJ
ejde-942	277	40	|λk	|λk	NUM
ejde-942	277	41	−	−	PROPN
ejde-942	277	42	λk|	λk|	NOUN
ejde-942	277	43	≤	≤	NUM
ejde-942	277	44	1	1	NUM
ejde-942	277	45	3	3	NUM
ejde-942	277	46	≤	≤	NUM
ejde-942	277	47	min	min	NOUN
ejde-942	277	48	j	j	PROPN
ejde-942	277	49	̸=k	̸=k	PROPN
ejde-942	277	50	{	{	PUNCT
ejde-942	277	51	|λk	|λk	PROPN
ejde-942	277	52	−	−	NOUN
ejde-942	277	53	λj	λj	PROPN
ejde-942	277	54	|	|	NOUN
ejde-942	277	55	}	}	PUNCT
ejde-942	277	56	.	.	PUNCT
ejde-942	278	1	by	by	ADP
ejde-942	278	2	a	a	DET
ejde-942	278	3	similar	similar	ADJ
ejde-942	278	4	argument	argument	NOUN
ejde-942	278	5	to	to	ADP
ejde-942	278	6	that	that	PRON
ejde-942	278	7	of	of	ADP
ejde-942	278	8	proposition	proposition	NOUN
ejde-942	278	9	3.4	3.4	NUM
ejde-942	278	10	,	,	PUNCT
ejde-942	278	11	we	we	PRON
ejde-942	278	12	can	can	AUX
ejde-942	278	13	prove	prove	VERB
ejde-942	278	14	proposition	proposition	NOUN
ejde-942	278	15	3.6	3.6	NUM
ejde-942	278	16	based	base	VERB
ejde-942	278	17	on	on	ADP
ejde-942	278	18	lemma	lemma	PROPN
ejde-942	278	19	3.3	3.3	NUM
ejde-942	278	20	.	.	PUNCT
ejde-942	279	1	denote	denote	VERB
ejde-942	279	2	ek(x	ek(x	NOUN
ejde-942	279	3	)	)	PUNCT
ejde-942	280	1	=	=	VERB
ejde-942	280	2	l[φθk	l[φθk	X
ejde-942	280	3	]	]	X
ejde-942	280	4	(	(	PUNCT
ejde-942	280	5	x)−λkφθk(x	x)−λkφθk(x	PROPN
ejde-942	280	6	)	)	PUNCT
ejde-942	280	7	,	,	PUNCT
ejde-942	280	8	we	we	PRON
ejde-942	280	9	can	can	AUX
ejde-942	280	10	estimate	estimate	VERB
ejde-942	280	11	the	the	DET
ejde-942	280	12	error	error	NOUN
ejde-942	280	13	of	of	ADP
ejde-942	280	14	the	the	DET
ejde-942	280	15	eigenvalue	eigenvalue	NOUN
ejde-942	280	16	by	by	ADP
ejde-942	280	17	∥ek(x)∥σ	∥ek(x)∥σ	PROPN
ejde-942	280	18	and	and	CCONJ
ejde-942	280	19	∥φθk∥σ	∥φθk∥σ	PROPN
ejde-942	280	20	.	.	PUNCT
ejde-942	281	1	theorem	theorem	VERB
ejde-942	281	2	3.7	3.7	NUM
ejde-942	281	3	(	(	PUNCT
ejde-942	281	4	eigenvalue	eigenvalue	NOUN
ejde-942	281	5	estimation	estimation	NOUN
ejde-942	281	6	)	)	PUNCT
ejde-942	281	7	.	.	PUNCT
ejde-942	282	1	if	if	SCONJ
ejde-942	282	2	|λk−λk|	|λk−λk|	VERB
ejde-942	282	3	≤	≤	NUM
ejde-942	282	4	minj	minj	NOUN
ejde-942	282	5	̸=k{|λk−λj	̸=k{|λk−λj	NOUN
ejde-942	282	6	|	|	NOUN
ejde-942	282	7	}	}	PUNCT
ejde-942	282	8	,	,	PUNCT
ejde-942	282	9	k	k	PROPN
ejde-942	282	10	∈	∈	PROPN
ejde-942	282	11	n+	n+	PROPN
ejde-942	282	12	,	,	PUNCT
ejde-942	282	13	we	we	PRON
ejde-942	282	14	have	have	VERB
ejde-942	282	15	|λk	|λk	NUM
ejde-942	282	16	−	−	PROPN
ejde-942	282	17	λk|	λk|	PROPN
ejde-942	282	18	≤	≤	NOUN
ejde-942	282	19	∥ek(x)∥σ	∥ek(x)∥σ	PROPN
ejde-942	282	20	∥φθk∥σ	∥φθk∥σ	PROPN
ejde-942	282	21	.	.	PUNCT
ejde-942	283	1	proof	proof	NOUN
ejde-942	283	2	.	.	PUNCT
ejde-942	284	1	note	note	VERB
ejde-942	284	2	that	that	SCONJ
ejde-942	284	3	∥l[φθk	∥l[φθk	PRON
ejde-942	284	4	]	]	SYM
ejde-942	284	5	−	−	X
ejde-942	284	6	λkφθk∥2σ	λkφθk∥2σ	NOUN
ejde-942	284	7	=	=	PUNCT
ejde-942	284	8	(	(	PUNCT
ejde-942	284	9	l[φθk	l[φθk	X
ejde-942	284	10	]	]	X
ejde-942	284	11	−	−	PROPN
ejde-942	284	12	λkφθk	λkφθk	NOUN
ejde-942	284	13	,	,	PUNCT
ejde-942	284	14	l[φθk	l[φθk	X
ejde-942	284	15	]	]	X
ejde-942	284	16	−	−	PROPN
ejde-942	284	17	λkφθk)σ	λkφθk)σ	PROPN
ejde-942	284	18	=	=	PUNCT
ejde-942	285	1	∞∑	∞∑	NUM
ejde-942	285	2	j=1	j=1	NOUN
ejde-942	285	3	(	(	PUNCT
ejde-942	285	4	λj	λj	PROPN
ejde-942	285	5	−	−	PROPN
ejde-942	285	6	λk	λk	NOUN
ejde-942	285	7	)	)	PUNCT
ejde-942	285	8	2|(φθk	2|(φθk	NUM
ejde-942	285	9	,	,	PUNCT
ejde-942	285	10	φj)σ|2	φj)σ|2	ADJ
ejde-942	285	11	ejde-2024/53	ejde-2024/53	NOUN
ejde-942	285	12	deep	deep	ADJ
ejde-942	285	13	learning	learning	NOUN
ejde-942	285	14	method	method	NOUN
ejde-942	285	15	for	for	ADP
ejde-942	285	16	finding	find	VERB
ejde-942	285	17	eigenpairs	eigenpair	NOUN
ejde-942	285	18	11	11	NUM
ejde-942	285	19	≥	≥	NOUN
ejde-942	285	20	(	(	PUNCT
ejde-942	285	21	λk	λk	ADP
ejde-942	285	22	−	−	PROPN
ejde-942	285	23	λk	λk	NOUN
ejde-942	285	24	)	)	PUNCT
ejde-942	285	25	2	2	NUM
ejde-942	285	26	∞∑	∞∑	NUM
ejde-942	285	27	j=1	j=1	NOUN
ejde-942	285	28	|(φθk	|(φθk	NUM
ejde-942	285	29	,	,	PUNCT
ejde-942	285	30	φj)σ|2	φj)σ|2	ADJ
ejde-942	285	31	thus	thus	ADV
ejde-942	285	32	,	,	PUNCT
ejde-942	285	33	we	we	PRON
ejde-942	285	34	conclude	conclude	VERB
ejde-942	285	35	that	that	SCONJ
ejde-942	285	36	|λk	|λk	NUM
ejde-942	286	1	−	−	PROPN
ejde-942	286	2	λk|	λk|	PROPN
ejde-942	286	3	≤	≤	NOUN
ejde-942	286	4	∥ek(x)∥σ	∥ek(x)∥σ	PROPN
ejde-942	286	5	∥φθk∥σ	∥φθk∥σ	PROPN
ejde-942	286	6	.	.	PUNCT
ejde-942	287	1	□	□	PUNCT
ejde-942	287	2	we	we	PRON
ejde-942	287	3	define	define	VERB
ejde-942	287	4	pkφ	pkφ	NOUN
ejde-942	287	5	=	=	SYM
ejde-942	287	6	(	(	PUNCT
ejde-942	287	7	φ	φ	PROPN
ejde-942	287	8	,	,	PUNCT
ejde-942	287	9	φk)σφk	φk)σφk	PROPN
ejde-942	287	10	and	and	CCONJ
ejde-942	287	11	p⊥	p⊥	PROPN
ejde-942	288	1	k	k	PROPN
ejde-942	288	2	φ	φ	PROPN
ejde-942	288	3	=	=	SYM
ejde-942	288	4	∑∞	∑∞	PROPN
ejde-942	288	5	j	j	PROPN
ejde-942	288	6	̸=k(φ	̸=k(φ	PROPN
ejde-942	288	7	,	,	PUNCT
ejde-942	288	8	φj)σφj	φj)σφj	PROPN
ejde-942	288	9	.	.	PUNCT
ejde-942	289	1	obviously	obviously	ADV
ejde-942	289	2	,	,	PUNCT
ejde-942	289	3	φθk	φθk	NOUN
ejde-942	289	4	=	=	NOUN
ejde-942	289	5	pkφθk	pkφθk	NOUN
ejde-942	289	6	+	+	CCONJ
ejde-942	289	7	p⊥	p⊥	NOUN
ejde-942	289	8	k	k	ADJ
ejde-942	289	9	φθk	φθk	NOUN
ejde-942	289	10	.	.	PUNCT
ejde-942	290	1	next	next	ADV
ejde-942	290	2	,	,	PUNCT
ejde-942	290	3	we	we	PRON
ejde-942	290	4	use	use	VERB
ejde-942	290	5	∥p⊥	∥p⊥	NOUN
ejde-942	290	6	k	k	PROPN
ejde-942	290	7	φθk	φθk	NOUN
ejde-942	290	8	|	|	ADV
ejde-942	290	9	to	to	PART
ejde-942	290	10	evaluate	evaluate	VERB
ejde-942	290	11	the	the	DET
ejde-942	290	12	eigenfunction	eigenfunction	NOUN
ejde-942	290	13	of	of	ADP
ejde-942	290	14	our	our	PRON
ejde-942	290	15	algorithm	algorithm	NOUN
ejde-942	290	16	.	.	PUNCT
ejde-942	291	1	theorem	theorem	VERB
ejde-942	291	2	3.8	3.8	NUM
ejde-942	291	3	(	(	PUNCT
ejde-942	291	4	eigenfunction	eigenfunction	NOUN
ejde-942	291	5	estimation	estimation	NOUN
ejde-942	291	6	)	)	PUNCT
ejde-942	291	7	.	.	PUNCT
ejde-942	292	1	assume	assume	VERB
ejde-942	292	2	that	that	SCONJ
ejde-942	292	3	ρ1	ρ1	NOUN
ejde-942	292	4	−	−	PROPN
ejde-942	293	1	ρ0	ρ0	PROPN
ejde-942	293	2	<	<	X
ejde-942	293	3	1	1	NUM
ejde-942	293	4	.	.	PUNCT
ejde-942	294	1	if	if	SCONJ
ejde-942	294	2	λk	λk	PROPN
ejde-942	294	3	∈	∈	PROPN
ejde-942	294	4	[	[	X
ejde-942	294	5	k2	k2	PROPN
ejde-942	294	6	+	+	CCONJ
ejde-942	294	7	ρ0	ρ0	PROPN
ejde-942	294	8	,	,	PUNCT
ejde-942	294	9	k	k	PROPN
ejde-942	294	10	2	2	NUM
ejde-942	294	11	+	+	NUM
ejde-942	294	12	ρ1	ρ1	NOUN
ejde-942	294	13	]	]	PUNCT
ejde-942	294	14	,	,	PUNCT
ejde-942	294	15	k	k	PROPN
ejde-942	294	16	∈	∈	PROPN
ejde-942	294	17	n+	n+	PROPN
ejde-942	294	18	,	,	PUNCT
ejde-942	294	19	the	the	DET
ejde-942	294	20	sturm	sturm	NOUN
ejde-942	294	21	-	-	PUNCT
ejde-942	294	22	liouville	liouville	NOUN
ejde-942	294	23	eigenvalue	eigenvalue	NOUN
ejde-942	294	24	problems	problem	NOUN
ejde-942	294	25	with	with	ADP
ejde-942	294	26	case	case	NOUN
ejde-942	294	27	(	(	PUNCT
ejde-942	294	28	1	1	X
ejde-942	294	29	)	)	PUNCT
ejde-942	294	30	satisfy	satisfy	NOUN
ejde-942	294	31	∥p⊥	∥p⊥	PROPN
ejde-942	294	32	1	1	NUM
ejde-942	294	33	φθ1∥σ	φθ1∥σ	PROPN
ejde-942	294	34	≤	≤	PROPN
ejde-942	294	35	∥e1(x)∥σ	∥e1(x)∥σ	VERB
ejde-942	294	36	|λ2	|λ2	NOUN
ejde-942	294	37	−	−	PUNCT
ejde-942	294	38	λ1|	λ1|	NOUN
ejde-942	294	39	,	,	PUNCT
ejde-942	294	40	or	or	CCONJ
ejde-942	294	41	∥p⊥	∥p⊥	PROPN
ejde-942	295	1	k	k	PROPN
ejde-942	295	2	φθk∥σ	φθk∥σ	PROPN
ejde-942	295	3	≤	≤	NUM
ejde-942	295	4	∥ek(x)∥σ	∥ek(x)∥σ	PROPN
ejde-942	295	5	min{|λk+1	min{|λk+1	VERB
ejde-942	295	6	−	−	PROPN
ejde-942	295	7	λk|	λk|	PROPN
ejde-942	295	8	,	,	PUNCT
ejde-942	295	9	|λk−1	|λk−1	NOUN
ejde-942	295	10	−	−	NOUN
ejde-942	295	11	λk|	λk|	NOUN
ejde-942	295	12	}	}	PUNCT
ejde-942	295	13	,	,	PUNCT
ejde-942	295	14	k	k	X
ejde-942	295	15	≥	≥	NUM
ejde-942	295	16	2	2	NUM
ejde-942	295	17	.	.	PUNCT
ejde-942	295	18	proof	proof	NOUN
ejde-942	295	19	.	.	PUNCT
ejde-942	296	1	by	by	ADP
ejde-942	296	2	proposition	proposition	NOUN
ejde-942	296	3	3.4	3.4	NUM
ejde-942	296	4	,	,	PUNCT
ejde-942	296	5	we	we	PRON
ejde-942	296	6	have	have	VERB
ejde-942	296	7	|λk	|λk	NUM
ejde-942	296	8	−	−	NOUN
ejde-942	297	1	λj	λj	INTJ
ejde-942	297	2	|	|	ADV
ejde-942	297	3	≥	≥	NOUN
ejde-942	297	4	1	1	NUM
ejde-942	297	5	,	,	PUNCT
ejde-942	297	6	k	k	PROPN
ejde-942	297	7	̸=	̸=	PROPN
ejde-942	297	8	j.	j.	PROPN
ejde-942	297	9	for	for	ADP
ejde-942	297	10	k	k	PROPN
ejde-942	297	11	=	=	SYM
ejde-942	297	12	1	1	NUM
ejde-942	297	13	,	,	PUNCT
ejde-942	297	14	we	we	PRON
ejde-942	297	15	obtain	obtain	VERB
ejde-942	297	16	∥l[φθ1	∥l[φθ1	NOUN
ejde-942	297	17	]	]	X
ejde-942	297	18	−	−	X
ejde-942	298	1	λ1φθ1∥2σ	λ1φθ1∥2σ	PUNCT
ejde-942	299	1	=	=	SYM
ejde-942	300	1	(	(	PUNCT
ejde-942	300	2	l[φθ1	l[φθ1	PROPN
ejde-942	300	3	]	]	X
ejde-942	300	4	−	−	PROPN
ejde-942	300	5	λ1φθ1	λ1φθ1	NOUN
ejde-942	300	6	,	,	PUNCT
ejde-942	300	7	l[φθ1	l[φθ1	PROPN
ejde-942	300	8	]	]	X
ejde-942	300	9	−	−	PROPN
ejde-942	301	1	λ1φθ1)σ	λ1φθ1)σ	NOUN
ejde-942	301	2	=	=	PUNCT
ejde-942	302	1	∞∑	∞∑	NUM
ejde-942	302	2	j=1	j=1	NOUN
ejde-942	302	3	(	(	PUNCT
ejde-942	302	4	λj	λj	PROPN
ejde-942	302	5	−	−	PROPN
ejde-942	302	6	λ1	λ1	PROPN
ejde-942	302	7	)	)	PUNCT
ejde-942	302	8	2|(φθ1	2|(φθ1	NOUN
ejde-942	302	9	,	,	PUNCT
ejde-942	302	10	φj)σ|2	φj)σ|2	X
ejde-942	302	11	≥	≥	X
ejde-942	302	12	(	(	PUNCT
ejde-942	302	13	λ2	λ2	NOUN
ejde-942	302	14	−	−	PROPN
ejde-942	302	15	λ1	λ1	PROPN
ejde-942	302	16	)	)	PUNCT
ejde-942	302	17	2	2	NUM
ejde-942	303	1	∞∑	∞∑	PROPN
ejde-942	303	2	j=2	j=2	PROPN
ejde-942	303	3	|(φθ1	|(φθ1	NOUN
ejde-942	303	4	,	,	PUNCT
ejde-942	303	5	φj)σ|2	φj)σ|2	ADJ
ejde-942	303	6	+	+	CCONJ
ejde-942	303	7	(	(	PUNCT
ejde-942	303	8	λ1	λ1	PROPN
ejde-942	303	9	−	−	PROPN
ejde-942	303	10	λ1	λ1	PROPN
ejde-942	303	11	)	)	PUNCT
ejde-942	303	12	2|(φθ1	2|(φθ1	NOUN
ejde-942	303	13	,	,	PUNCT
ejde-942	303	14	φ1)σ|2	φ1)σ|2	X
ejde-942	303	15	≥	≥	X
ejde-942	303	16	(	(	PUNCT
ejde-942	303	17	λ2	λ2	NOUN
ejde-942	303	18	−	−	PROPN
ejde-942	303	19	λ1	λ1	PROPN
ejde-942	303	20	)	)	PUNCT
ejde-942	303	21	2	2	NUM
ejde-942	304	1	∞∑	∞∑	PROPN
ejde-942	304	2	j=2	j=2	PROPN
ejde-942	304	3	|(φθ1	|(φθ1	NOUN
ejde-942	304	4	,	,	PUNCT
ejde-942	304	5	φj)σ|2	φj)σ|2	ADJ
ejde-942	304	6	thus	thus	ADV
ejde-942	304	7	,	,	PUNCT
ejde-942	304	8	we	we	PRON
ejde-942	304	9	conclude	conclude	VERB
ejde-942	304	10	that	that	DET
ejde-942	304	11	∥p⊥	∥p⊥	NOUN
ejde-942	304	12	1	1	NUM
ejde-942	304	13	φθ1∥σ	φθ1∥σ	PROPN
ejde-942	304	14	≤	≤	PROPN
ejde-942	304	15	∥e1(x)∥σ	∥e1(x)∥σ	VERB
ejde-942	304	16	|λ2	|λ2	NOUN
ejde-942	304	17	−	−	PUNCT
ejde-942	304	18	λ1|	λ1|	PROPN
ejde-942	304	19	for	for	ADP
ejde-942	304	20	k	k	PROPN
ejde-942	304	21	≥	≥	NUM
ejde-942	304	22	2	2	NUM
ejde-942	304	23	,	,	PUNCT
ejde-942	304	24	we	we	PRON
ejde-942	304	25	obtain	obtain	VERB
ejde-942	304	26	∥l[φθk	∥l[φθk	PUNCT
ejde-942	304	27	]	]	PUNCT
ejde-942	304	28	−	−	X
ejde-942	305	1	λkφθk∥2σ	λkφθk∥2σ	NOUN
ejde-942	305	2	=	=	PUNCT
ejde-942	305	3	(	(	PUNCT
ejde-942	305	4	l[φθk	l[φθk	X
ejde-942	305	5	]	]	X
ejde-942	305	6	−	−	PROPN
ejde-942	305	7	λkφθk	λkφθk	NOUN
ejde-942	305	8	,	,	PUNCT
ejde-942	305	9	l[φθk	l[φθk	X
ejde-942	305	10	]	]	X
ejde-942	305	11	−	−	PROPN
ejde-942	305	12	λkφθk)σ	λkφθk)σ	PROPN
ejde-942	305	13	=	=	PUNCT
ejde-942	306	1	∞∑	∞∑	NUM
ejde-942	306	2	j=1	j=1	NOUN
ejde-942	306	3	(	(	PUNCT
ejde-942	306	4	λj	λj	PROPN
ejde-942	306	5	−	−	PROPN
ejde-942	306	6	λk	λk	NOUN
ejde-942	306	7	)	)	PUNCT
ejde-942	306	8	2|(φθk	2|(φθk	NUM
ejde-942	306	9	,	,	PUNCT
ejde-942	306	10	φj)σ|2	φj)σ|2	ADJ
ejde-942	306	11	≥	≥	NOUN
ejde-942	306	12	min{(λk+1	min{(λk+1	VERB
ejde-942	306	13	−	−	PROPN
ejde-942	306	14	λk	λk	NOUN
ejde-942	306	15	)	)	PUNCT
ejde-942	306	16	2	2	NUM
ejde-942	306	17	,	,	PUNCT
ejde-942	306	18	(	(	PUNCT
ejde-942	306	19	λk−1	λk−1	NOUN
ejde-942	306	20	−	−	NOUN
ejde-942	306	21	λk	λk	NOUN
ejde-942	306	22	)	)	PUNCT
ejde-942	306	23	2	2	NUM
ejde-942	306	24	}	}	PUNCT
ejde-942	306	25	∞∑	∞∑	NUM
ejde-942	306	26	j=1,j	j=1,j	PROPN
ejde-942	306	27	̸=k	̸=k	PROPN
ejde-942	306	28	|(φθk	|(φθk	NUM
ejde-942	306	29	,	,	PUNCT
ejde-942	306	30	φj)σ|2	φj)σ|2	ADJ
ejde-942	306	31	thus	thus	ADV
ejde-942	306	32	,	,	PUNCT
ejde-942	306	33	we	we	PRON
ejde-942	306	34	conclude	conclude	VERB
ejde-942	306	35	that	that	DET
ejde-942	306	36	∥p⊥	∥p⊥	NOUN
ejde-942	306	37	k	k	PROPN
ejde-942	306	38	φθk∥σ	φθk∥σ	PROPN
ejde-942	306	39	≤	≤	NUM
ejde-942	306	40	∥ek(x)∥σ	∥ek(x)∥σ	PROPN
ejde-942	306	41	min{|λk+1	min{|λk+1	VERB
ejde-942	306	42	−	−	PROPN
ejde-942	306	43	λk|	λk|	PROPN
ejde-942	306	44	,	,	PUNCT
ejde-942	306	45	|λk−1	|λk−1	NOUN
ejde-942	306	46	−	−	NOUN
ejde-942	307	1	λk|	λk|	NOUN
ejde-942	307	2	}	}	PUNCT
ejde-942	307	3	□	□	PUNCT
ejde-942	307	4	remark	remark	NOUN
ejde-942	307	5	3.9	3.9	NUM
ejde-942	307	6	.	.	PUNCT
ejde-942	308	1	by	by	ADP
ejde-942	308	2	proposition3.5	proposition3.5	NOUN
ejde-942	308	3	and	and	CCONJ
ejde-942	308	4	3.6	3.6	NUM
ejde-942	308	5	,	,	PUNCT
ejde-942	308	6	theorem	theorem	VERB
ejde-942	308	7	3.8	3.8	NUM
ejde-942	308	8	can	can	AUX
ejde-942	308	9	be	be	AUX
ejde-942	308	10	easily	easily	ADV
ejde-942	308	11	extended	extend	VERB
ejde-942	308	12	to	to	ADP
ejde-942	308	13	:	:	PUNCT
ejde-942	308	14	12	12	NUM
ejde-942	308	15	s.	s.	PROPN
ejde-942	308	16	zhang	zhang	PROPN
ejde-942	308	17	,	,	PUNCT
ejde-942	308	18	j.	j.	PROPN
ejde-942	308	19	zu	zu	PROPN
ejde-942	308	20	,	,	PUNCT
ejde-942	308	21	j.	j.	PROPN
ejde-942	308	22	zhang	zhang	PROPN
ejde-942	308	23	ejde-2024/53	ejde-2024/53	PROPN
ejde-942	308	24	(	(	PUNCT
ejde-942	308	25	1	1	NUM
ejde-942	308	26	)	)	PUNCT
ejde-942	308	27	sturm	sturm	NOUN
ejde-942	308	28	-	-	PUNCT
ejde-942	308	29	liouville	liouville	NOUN
ejde-942	308	30	eigenvalue	eigenvalue	NOUN
ejde-942	308	31	problems	problem	NOUN
ejde-942	308	32	with	with	ADP
ejde-942	308	33	case	case	NOUN
ejde-942	308	34	(	(	PUNCT
ejde-942	308	35	3	3	NUM
ejde-942	308	36	)	)	PUNCT
ejde-942	308	37	and	and	CCONJ
ejde-942	308	38	(	(	PUNCT
ejde-942	308	39	4	4	NUM
ejde-942	308	40	)	)	PUNCT
ejde-942	308	41	,	,	PUNCT
ejde-942	308	42	if	if	SCONJ
ejde-942	308	43	a2	a2	PROPN
ejde-942	308	44	=	=	SYM
ejde-942	308	45	0	0	NUM
ejde-942	308	46	,	,	PUNCT
ejde-942	308	47	α1	α1	X
ejde-942	308	48	≥	≥	NUM
ejde-942	308	49	0	0	NUM
ejde-942	308	50	,	,	PUNCT
ejde-942	308	51	ρ1−ρ0	ρ1−ρ0	NUM
ejde-942	308	52	+	+	SYM
ejde-942	308	53	2α1	2α1	NUM
ejde-942	308	54	π	π	NOUN
ejde-942	308	55	≤	≤	NUM
ejde-942	308	56	2	2	NUM
ejde-942	308	57	3	3	NUM
ejde-942	308	58	and	and	CCONJ
ejde-942	308	59	λk	λk	ADP
ejde-942	308	60	∈	∈	PROPN
ejde-942	308	61	[	[	PUNCT
ejde-942	308	62	(	(	PUNCT
ejde-942	308	63	k	k	NOUN
ejde-942	308	64	−	−	PROPN
ejde-942	308	65	1	1	NUM
ejde-942	308	66	2	2	NUM
ejde-942	308	67	)	)	PUNCT
ejde-942	308	68	2	2	NUM
ejde-942	309	1	+	+	NOUN
ejde-942	309	2	ρ0	ρ0	PROPN
ejde-942	309	3	,	,	PUNCT
ejde-942	309	4	(	(	PUNCT
ejde-942	309	5	k	k	NOUN
ejde-942	309	6	−	−	PROPN
ejde-942	309	7	1	1	NUM
ejde-942	309	8	2	2	NUM
ejde-942	309	9	)	)	PUNCT
ejde-942	309	10	2	2	NUM
ejde-942	309	11	+	+	NOUN
ejde-942	309	12	ρ1	ρ1	NOUN
ejde-942	309	13	+	+	SYM
ejde-942	309	14	2α1	2α1	NUM
ejde-942	309	15	π	π	NOUN
ejde-942	309	16	]	]	X
ejde-942	309	17	(	(	PUNCT
ejde-942	309	18	resp	resp	NOUN
ejde-942	309	19	.	.	PUNCT
ejde-942	310	1	a1	a1	NOUN
ejde-942	310	2	=	=	SYM
ejde-942	310	3	0	0	PROPN
ejde-942	310	4	,	,	PUNCT
ejde-942	310	5	α2	α2	ADJ
ejde-942	310	6	≥	≥	NOUN
ejde-942	310	7	0	0	NUM
ejde-942	310	8	,	,	PUNCT
ejde-942	310	9	ρ1−ρ0	ρ1−ρ0	NUM
ejde-942	310	10	+	+	SYM
ejde-942	310	11	2α2	2α2	NUM
ejde-942	310	12	π	π	NOUN
ejde-942	310	13	≤	≤	NUM
ejde-942	310	14	2	2	NUM
ejde-942	310	15	3	3	NUM
ejde-942	310	16	and	and	CCONJ
ejde-942	310	17	λk	λk	ADP
ejde-942	310	18	∈	∈	PROPN
ejde-942	310	19	[	[	PUNCT
ejde-942	310	20	(	(	PUNCT
ejde-942	310	21	k	k	NOUN
ejde-942	310	22	−	−	PROPN
ejde-942	310	23	1	1	NUM
ejde-942	310	24	2	2	NUM
ejde-942	310	25	)	)	PUNCT
ejde-942	310	26	2	2	NUM
ejde-942	311	1	+	+	NOUN
ejde-942	311	2	ρ0	ρ0	PROPN
ejde-942	311	3	,	,	PUNCT
ejde-942	311	4	(	(	PUNCT
ejde-942	311	5	k	k	NOUN
ejde-942	311	6	−	−	PROPN
ejde-942	311	7	1	1	NUM
ejde-942	311	8	2	2	NUM
ejde-942	311	9	)	)	PUNCT
ejde-942	311	10	2	2	NUM
ejde-942	311	11	+	+	NOUN
ejde-942	311	12	ρ1	ρ1	NOUN
ejde-942	311	13	+	+	X
ejde-942	311	14	2α2	2α2	NUM
ejde-942	311	15	π	π	NOUN
ejde-942	311	16	]	]	X
ejde-942	311	17	)	)	PUNCT
ejde-942	311	18	.	.	PUNCT
ejde-942	312	1	(	(	PUNCT
ejde-942	312	2	2	2	X
ejde-942	312	3	)	)	PUNCT
ejde-942	312	4	sturm	sturm	NOUN
ejde-942	312	5	-	-	PUNCT
ejde-942	312	6	liouville	liouville	NOUN
ejde-942	312	7	eigenvalue	eigenvalue	NOUN
ejde-942	312	8	problems	problem	NOUN
ejde-942	312	9	with	with	ADP
ejde-942	312	10	case	case	NOUN
ejde-942	312	11	(	(	PUNCT
ejde-942	312	12	2	2	NUM
ejde-942	312	13	)	)	PUNCT
ejde-942	312	14	,	,	PUNCT
ejde-942	312	15	(	(	PUNCT
ejde-942	312	16	5	5	NUM
ejde-942	312	17	)	)	PUNCT
ejde-942	312	18	and	and	CCONJ
ejde-942	312	19	(	(	PUNCT
ejde-942	312	20	6	6	NUM
ejde-942	312	21	)	)	PUNCT
ejde-942	312	22	,	,	PUNCT
ejde-942	312	23	if	if	SCONJ
ejde-942	312	24	α1	α1	PROPN
ejde-942	312	25	≥	≥	NUM
ejde-942	312	26	0	0	NUM
ejde-942	312	27	,	,	PUNCT
ejde-942	312	28	α2	α2	PROPN
ejde-942	312	29	≥	≥	NOUN
ejde-942	312	30	0	0	NUM
ejde-942	312	31	,	,	PUNCT
ejde-942	312	32	ρ1	ρ1	NOUN
ejde-942	312	33	−	−	PROPN
ejde-942	312	34	ρ0	ρ0	PROPN
ejde-942	313	1	+	+	NOUN
ejde-942	313	2	2α1	2α1	NUM
ejde-942	313	3	π	π	X
ejde-942	313	4	+	+	CCONJ
ejde-942	313	5	2α2	2α2	NUM
ejde-942	313	6	π	π	PROPN
ejde-942	313	7	≤	≤	NUM
ejde-942	313	8	1	1	NUM
ejde-942	313	9	3	3	NUM
ejde-942	313	10	and	and	CCONJ
ejde-942	313	11	λk	λk	ADP
ejde-942	313	12	∈	∈	PROPN
ejde-942	314	1	[	[	X
ejde-942	314	2	(	(	PUNCT
ejde-942	314	3	k	k	PROPN
ejde-942	314	4	−	−	PROPN
ejde-942	314	5	1)2	1)2	NUM
ejde-942	314	6	+	+	CCONJ
ejde-942	314	7	ρ0	ρ0	PROPN
ejde-942	314	8	,	,	PUNCT
ejde-942	314	9	(	(	PUNCT
ejde-942	314	10	k	k	PROPN
ejde-942	314	11	−	−	PROPN
ejde-942	314	12	1)2	1)2	NUM
ejde-942	314	13	+	+	NUM
ejde-942	314	14	ρ1	ρ1	NOUN
ejde-942	314	15	+	+	CCONJ
ejde-942	314	16	2α1	2α1	NUM
ejde-942	314	17	π	π	X
ejde-942	314	18	+	+	CCONJ
ejde-942	314	19	2α2	2α2	NUM
ejde-942	314	20	π	π	NOUN
ejde-942	314	21	]	]	X
ejde-942	314	22	.	.	PUNCT
ejde-942	315	1	4	4	X
ejde-942	315	2	.	.	X
ejde-942	315	3	comparison	comparison	NOUN
ejde-942	315	4	with	with	ADP
ejde-942	315	5	previous	previous	ADJ
ejde-942	315	6	algorithms	algorithm	NOUN
ejde-942	315	7	4.1	4.1	NUM
ejde-942	315	8	.	.	PUNCT
ejde-942	316	1	comparison	comparison	NOUN
ejde-942	316	2	with	with	ADP
ejde-942	316	3	the	the	DET
ejde-942	316	4	state	state	NOUN
ejde-942	316	5	-	-	PUNCT
ejde-942	316	6	of	of	ADP
ejde-942	316	7	-	-	PUNCT
ejde-942	316	8	the	the	DET
ejde-942	316	9	-	-	PUNCT
ejde-942	316	10	art	art	NOUN
ejde-942	316	11	deep	deep	ADJ
ejde-942	316	12	learning	learning	NOUN
ejde-942	316	13	methods	method	NOUN
ejde-942	316	14	.	.	PUNCT
ejde-942	317	1	consider	consider	VERB
ejde-942	317	2	the	the	DET
ejde-942	317	3	smallest	small	ADJ
ejde-942	317	4	eigenpairs	eigenpair	NOUN
ejde-942	317	5	problems	problem	NOUN
ejde-942	317	6	φ′′(x	φ′′(x	NOUN
ejde-942	317	7	)	)	PUNCT
ejde-942	317	8	+	+	NUM
ejde-942	317	9	λφ(x	λφ(x	X
ejde-942	317	10	)	)	PUNCT
ejde-942	317	11	=	=	SYM
ejde-942	317	12	0	0	NUM
ejde-942	317	13	,	,	PUNCT
ejde-942	317	14	(	(	PUNCT
ejde-942	317	15	4.1	4.1	NUM
ejde-942	317	16	)	)	PUNCT
ejde-942	317	17	with	with	ADP
ejde-942	317	18	dirichlet	dirichlet	PROPN
ejde-942	317	19	boundary	boundary	PROPN
ejde-942	317	20	value	value	NOUN
ejde-942	317	21	conditions	condition	NOUN
ejde-942	317	22	φ(0	φ(0	ADJ
ejde-942	317	23	)	)	PUNCT
ejde-942	317	24	=	=	SYM
ejde-942	317	25	φ(π	φ(π	NOUN
ejde-942	317	26	)	)	PUNCT
ejde-942	317	27	=	=	SYM
ejde-942	318	1	0	0	X
ejde-942	318	2	.	.	PUNCT
ejde-942	318	3	(	(	PUNCT
ejde-942	318	4	4.2	4.2	NUM
ejde-942	318	5	)	)	PUNCT
ejde-942	318	6	ben	ben	PROPN
ejde-942	318	7	-	-	PUNCT
ejde-942	318	8	shaul	shaul	PROPN
ejde-942	318	9	et	et	NOUN
ejde-942	318	10	al	al	PROPN
ejde-942	318	11	.	.	PUNCT
ejde-942	319	1	[	[	X
ejde-942	319	2	5	5	NUM
ejde-942	319	3	]	]	PUNCT
ejde-942	319	4	utilized	utilize	VERB
ejde-942	319	5	a	a	DET
ejde-942	319	6	more	more	ADV
ejde-942	319	7	intricate	intricate	ADJ
ejde-942	319	8	cost	cost	NOUN
ejde-942	319	9	function	function	NOUN
ejde-942	319	10	to	to	PART
ejde-942	319	11	simultaneously	simultaneously	ADV
ejde-942	319	12	identify	identify	VERB
ejde-942	319	13	the	the	DET
ejde-942	319	14	four	four	NUM
ejde-942	319	15	smallest	small	ADJ
ejde-942	319	16	eigenpairs	eigenpair	NOUN
ejde-942	319	17	.	.	PUNCT
ejde-942	320	1	in	in	ADP
ejde-942	320	2	table	table	NOUN
ejde-942	320	3	4.1	4.1	NUM
ejde-942	320	4	,	,	PUNCT
ejde-942	320	5	ben	ben	PROPN
ejde-942	320	6	-	-	PUNCT
ejde-942	320	7	shaul1	shaul1	PROPN
ejde-942	320	8	denotes	denote	VERB
ejde-942	320	9	the	the	DET
ejde-942	320	10	simultaneous	simultaneous	ADJ
ejde-942	320	11	training	training	NOUN
ejde-942	320	12	of	of	ADP
ejde-942	320	13	six	six	NUM
ejde-942	320	14	smallest	small	ADJ
ejde-942	320	15	eigenpairs	eigenpair	NOUN
ejde-942	320	16	using	use	VERB
ejde-942	320	17	60000	60000	NUM
ejde-942	320	18	epochs	epoch	NOUN
ejde-942	320	19	with	with	ADP
ejde-942	320	20	their	their	PRON
ejde-942	320	21	approach	approach	NOUN
ejde-942	320	22	.	.	PUNCT
ejde-942	321	1	benshaul2	benshaul2	PROPN
ejde-942	321	2	represents	represent	VERB
ejde-942	321	3	the	the	DET
ejde-942	321	4	concurrent	concurrent	ADJ
ejde-942	321	5	training	training	NOUN
ejde-942	321	6	of	of	ADP
ejde-942	321	7	seven	seven	NUM
ejde-942	321	8	smallest	small	ADJ
ejde-942	321	9	eigenpairs	eigenpair	NOUN
ejde-942	321	10	using	use	VERB
ejde-942	321	11	70000	70000	NUM
ejde-942	321	12	epochs	epoch	NOUN
ejde-942	321	13	.	.	PUNCT
ejde-942	322	1	we	we	PRON
ejde-942	322	2	find	find	VERB
ejde-942	322	3	that	that	SCONJ
ejde-942	322	4	their	their	PRON
ejde-942	322	5	algorithm	algorithm	NOUN
ejde-942	322	6	has	have	VERB
ejde-942	322	7	a	a	DET
ejde-942	322	8	large	large	ADJ
ejde-942	322	9	error	error	NOUN
ejde-942	322	10	when	when	SCONJ
ejde-942	322	11	n	n	X
ejde-942	322	12	=	=	SYM
ejde-942	322	13	6	6	NUM
ejde-942	322	14	,	,	PUNCT
ejde-942	322	15	and	and	CCONJ
ejde-942	322	16	they	they	PRON
ejde-942	322	17	obtained	obtain	VERB
ejde-942	322	18	incorrect	incorrect	ADJ
ejde-942	322	19	results	result	NOUN
ejde-942	322	20	when	when	SCONJ
ejde-942	322	21	n	n	X
ejde-942	322	22	=	=	SYM
ejde-942	322	23	7	7	X
ejde-942	322	24	.	.	PUNCT
ejde-942	323	1	in	in	ADP
ejde-942	323	2	contrast	contrast	NOUN
ejde-942	323	3	,	,	PUNCT
ejde-942	323	4	our	our	PRON
ejde-942	323	5	objective	objective	NOUN
ejde-942	323	6	differs	differ	VERB
ejde-942	323	7	significantly	significantly	ADV
ejde-942	323	8	.	.	PUNCT
ejde-942	324	1	our	our	PRON
ejde-942	324	2	aim	aim	NOUN
ejde-942	324	3	is	be	AUX
ejde-942	324	4	to	to	PART
ejde-942	324	5	devise	devise	VERB
ejde-942	324	6	an	an	DET
ejde-942	324	7	easy	easy	ADJ
ejde-942	324	8	-	-	PUNCT
ejde-942	324	9	to	to	PART
ejde-942	324	10	-	-	PUNCT
ejde-942	324	11	understand	understand	VERB
ejde-942	324	12	algorithm	algorithm	NOUN
ejde-942	324	13	and	and	CCONJ
ejde-942	324	14	adapt	adapt	VERB
ejde-942	324	15	it	it	PRON
ejde-942	324	16	for	for	ADP
ejde-942	324	17	handling	handle	VERB
ejde-942	324	18	larger	large	ADJ
ejde-942	324	19	eigenvalues	eigenvalue	NOUN
ejde-942	324	20	.	.	PUNCT
ejde-942	325	1	our	our	PRON
ejde-942	325	2	method	method	NOUN
ejde-942	325	3	involves	involve	VERB
ejde-942	325	4	sequentially	sequentially	ADV
ejde-942	325	5	training	train	VERB
ejde-942	325	6	10	10	NUM
ejde-942	325	7	smallest	small	ADJ
ejde-942	325	8	eigenpairs	eigenpair	NOUN
ejde-942	325	9	,	,	PUNCT
ejde-942	325	10	with	with	SCONJ
ejde-942	325	11	each	each	DET
ejde-942	325	12	pair	pair	NOUN
ejde-942	325	13	trained	train	VERB
ejde-942	325	14	using	use	VERB
ejde-942	325	15	10000	10000	NUM
ejde-942	325	16	epochs	epoch	NOUN
ejde-942	325	17	.	.	PUNCT
ejde-942	326	1	note	note	VERB
ejde-942	326	2	that	that	SCONJ
ejde-942	326	3	it	it	PRON
ejde-942	326	4	is	be	AUX
ejde-942	326	5	a	a	DET
ejde-942	326	6	fair	fair	ADJ
ejde-942	326	7	comparison	comparison	NOUN
ejde-942	326	8	,	,	PUNCT
ejde-942	326	9	the	the	DET
ejde-942	326	10	total	total	ADJ
ejde-942	326	11	number	number	NOUN
ejde-942	326	12	of	of	ADP
ejde-942	326	13	epochs	epoch	NOUN
ejde-942	326	14	used	use	VERB
ejde-942	326	15	in	in	ADP
ejde-942	326	16	our	our	PRON
ejde-942	326	17	method	method	NOUN
ejde-942	326	18	is	be	AUX
ejde-942	326	19	equivalent	equivalent	ADJ
ejde-942	326	20	to	to	ADP
ejde-942	326	21	that	that	PRON
ejde-942	326	22	of	of	ADP
ejde-942	326	23	the	the	DET
ejde-942	326	24	other	other	ADJ
ejde-942	326	25	methods	method	NOUN
ejde-942	326	26	.	.	PUNCT
ejde-942	327	1	we	we	PRON
ejde-942	327	2	set	set	VERB
ejde-942	327	3	the	the	DET
ejde-942	327	4	integral	integral	ADJ
ejde-942	327	5	point	point	NOUN
ejde-942	327	6	to	to	ADP
ejde-942	327	7	500	500	NUM
ejde-942	327	8	and	and	CCONJ
ejde-942	327	9	lr	lr	X
ejde-942	327	10	=	=	NOUN
ejde-942	327	11	0.01	0.01	NUM
ejde-942	327	12	.	.	PUNCT
ejde-942	328	1	the	the	DET
ejde-942	328	2	hyper	hyper	NOUN
ejde-942	328	3	-	-	NOUN
ejde-942	328	4	parameters	parameter	NOUN
ejde-942	328	5	are	be	AUX
ejde-942	328	6	set	set	VERB
ejde-942	328	7	as	as	ADP
ejde-942	328	8	in	in	ADP
ejde-942	328	9	[	[	X
ejde-942	328	10	5	5	NUM
ejde-942	328	11	]	]	PUNCT
ejde-942	328	12	.	.	PUNCT
ejde-942	329	1	we	we	PRON
ejde-942	329	2	found	find	VERB
ejde-942	329	3	that	that	SCONJ
ejde-942	329	4	the	the	DET
ejde-942	329	5	runtime	runtime	NOUN
ejde-942	329	6	of	of	ADP
ejde-942	329	7	our	our	PRON
ejde-942	329	8	method	method	NOUN
ejde-942	329	9	is	be	AUX
ejde-942	329	10	less	less	ADJ
ejde-942	329	11	than	than	ADP
ejde-942	329	12	both	both	CCONJ
ejde-942	329	13	benshaul1	benshaul1	PROPN
ejde-942	329	14	and	and	CCONJ
ejde-942	329	15	ben	ben	PROPN
ejde-942	329	16	-	-	PUNCT
ejde-942	329	17	shaul2	shaul2	PROPN
ejde-942	329	18	,	,	PUNCT
ejde-942	329	19	but	but	CCONJ
ejde-942	329	20	it	it	PRON
ejde-942	329	21	obviously	obviously	ADV
ejde-942	329	22	has	have	VERB
ejde-942	329	23	more	more	ADJ
ejde-942	329	24	accuracy	accuracy	NOUN
ejde-942	329	25	,	,	PUNCT
ejde-942	329	26	see	see	VERB
ejde-942	329	27	table	table	NOUN
ejde-942	329	28	4.1	4.1	NUM
ejde-942	329	29	.	.	PUNCT
ejde-942	330	1	for	for	ADP
ejde-942	330	2	our	our	PRON
ejde-942	330	3	method	method	NOUN
ejde-942	330	4	,	,	PUNCT
ejde-942	330	5	if	if	SCONJ
ejde-942	330	6	we	we	PRON
ejde-942	330	7	set	set	VERB
ejde-942	330	8	30000	30000	NUM
ejde-942	330	9	epochs	epoch	NOUN
ejde-942	330	10	for	for	ADP
ejde-942	330	11	each	each	DET
ejde-942	330	12	,	,	PUNCT
ejde-942	330	13	1000	1000	NUM
ejde-942	330	14	inner	inner	ADJ
ejde-942	330	15	points	point	NOUN
ejde-942	330	16	and	and	CCONJ
ejde-942	330	17	200	200	NUM
ejde-942	330	18	integral	integral	ADJ
ejde-942	330	19	points	point	NOUN
ejde-942	330	20	,	,	PUNCT
ejde-942	330	21	we	we	PRON
ejde-942	330	22	will	will	AUX
ejde-942	330	23	get	get	VERB
ejde-942	330	24	more	more	ADV
ejde-942	330	25	accurate	accurate	ADJ
ejde-942	330	26	eigenpairs	eigenpair	NOUN
ejde-942	330	27	.	.	PUNCT
ejde-942	331	1	table	table	NOUN
ejde-942	331	2	4.1	4.1	NUM
ejde-942	331	3	.	.	PUNCT
ejde-942	332	1	comparison	comparison	NOUN
ejde-942	332	2	with	with	ADP
ejde-942	332	3	ben	ben	PROPN
ejde-942	332	4	-	-	PUNCT
ejde-942	332	5	shaul	shaul	PROPN
ejde-942	332	6	et	et	PROPN
ejde-942	332	7	al	al	PROPN
ejde-942	332	8	.	.	PROPN
ejde-942	332	9	’s	’s	PART
ejde-942	332	10	method	method	NOUN
ejde-942	332	11	.	.	PUNCT
ejde-942	333	1	exact	exact	ADJ
ejde-942	333	2	value	value	PROPN
ejde-942	333	3	ben	ben	PROPN
ejde-942	333	4	-	-	PROPN
ejde-942	333	5	shaul1	shaul1	PROPN
ejde-942	333	6	ben	ben	PROPN
ejde-942	333	7	-	-	PUNCT
ejde-942	333	8	shaul2	shaul2	VERB
ejde-942	334	1	ours	ours	ADJ
ejde-942	334	2	(	(	PUNCT
ejde-942	334	3	runtime	runtime	NOUN
ejde-942	334	4	)	)	PUNCT
ejde-942	334	5	λ1	λ1	PROPN
ejde-942	334	6	=	=	NOUN
ejde-942	334	7	1	1	NUM
ejde-942	334	8	0.9859	0.9859	NUM
ejde-942	334	9	0.8639	0.8639	NUM
ejde-942	334	10	1.0000	1.0000	NUM
ejde-942	334	11	(	(	PUNCT
ejde-942	334	12	39s	39	NOUN
ejde-942	334	13	)	)	PUNCT
ejde-942	334	14	λ2	λ2	NOUN
ejde-942	334	15	=	=	SYM
ejde-942	334	16	4	4	NUM
ejde-942	334	17	3.9042	3.9042	NUM
ejde-942	334	18	1.0964	1.0964	NUM
ejde-942	334	19	4.0000	4.0000	NUM
ejde-942	334	20	(	(	PUNCT
ejde-942	334	21	81s	81	NOUN
ejde-942	334	22	)	)	PUNCT
ejde-942	335	1	λ3	λ3	PROPN
ejde-942	335	2	=	=	NOUN
ejde-942	335	3	9	9	NUM
ejde-942	335	4	8.8453	8.8453	NUM
ejde-942	335	5	4.6499	4.6499	NUM
ejde-942	335	6	8.9943	8.9943	NUM
ejde-942	335	7	(	(	PUNCT
ejde-942	335	8	149s	149	NOUN
ejde-942	335	9	)	)	PUNCT
ejde-942	335	10	λ4	λ4	PROPN
ejde-942	335	11	=	=	NOUN
ejde-942	335	12	16	16	NUM
ejde-942	335	13	15.4377	15.4377	NUM
ejde-942	335	14	8.5765	8.5765	NUM
ejde-942	335	15	16.0017	16.0017	NUM
ejde-942	335	16	(	(	PUNCT
ejde-942	335	17	270s	270	NOUN
ejde-942	335	18	)	)	PUNCT
ejde-942	335	19	λ5	λ5	NOUN
ejde-942	335	20	=	=	NOUN
ejde-942	335	21	25	25	NUM
ejde-942	335	22	24.4220	24.4220	NUM
ejde-942	335	23	16.2587	16.2587	NUM
ejde-942	335	24	24.9451	24.9451	NUM
ejde-942	335	25	(	(	PUNCT
ejde-942	335	26	338s	338s	PROPN
ejde-942	335	27	)	)	PUNCT
ejde-942	335	28	λ6	λ6	NOUN
ejde-942	335	29	=	=	NOUN
ejde-942	335	30	36	36	NUM
ejde-942	335	31	34.8055(1485s	34.8055(1485s	NUM
ejde-942	335	32	)	)	PUNCT
ejde-942	335	33	23.5614	23.5614	NUM
ejde-942	335	34	35.9928	35.9928	NUM
ejde-942	335	35	(	(	PUNCT
ejde-942	335	36	404s	404s	PROPN
ejde-942	335	37	)	)	PUNCT
ejde-942	335	38	λ7	λ7	NOUN
ejde-942	336	1	=	=	SYM
ejde-942	336	2	49	49	NUM
ejde-942	336	3	n	n	CCONJ
ejde-942	336	4	/	/	SYM
ejde-942	336	5	a	a	DET
ejde-942	336	6	35.5628(2128s	35.5628(2128s	NUM
ejde-942	336	7	)	)	PUNCT
ejde-942	336	8	48.9332	48.9332	NUM
ejde-942	336	9	(	(	PUNCT
ejde-942	336	10	475s	475s	NOUN
ejde-942	336	11	)	)	PUNCT
ejde-942	336	12	λ8	λ8	NOUN
ejde-942	336	13	=	=	PUNCT
ejde-942	336	14	64	64	NUM
ejde-942	336	15	n	n	NOUN
ejde-942	336	16	/	/	SYM
ejde-942	336	17	a	a	DET
ejde-942	336	18	n	n	NOUN
ejde-942	336	19	/	/	SYM
ejde-942	336	20	a	a	DET
ejde-942	336	21	63.9604	63.9604	NUM
ejde-942	336	22	(	(	PUNCT
ejde-942	336	23	552s	552s	PROPN
ejde-942	336	24	)	)	PUNCT
ejde-942	336	25	λ9	λ9	NOUN
ejde-942	336	26	=	=	SYM
ejde-942	336	27	81	81	NUM
ejde-942	336	28	n	n	CCONJ
ejde-942	336	29	/	/	SYM
ejde-942	336	30	a	a	DET
ejde-942	336	31	n	n	NOUN
ejde-942	336	32	/	/	SYM
ejde-942	336	33	a	a	DET
ejde-942	336	34	80.8455	80.8455	NUM
ejde-942	336	35	(	(	PUNCT
ejde-942	336	36	636s	636s	PROPN
ejde-942	336	37	)	)	PUNCT
ejde-942	336	38	λ10	λ10	NOUN
ejde-942	336	39	=	=	SYM
ejde-942	336	40	100	100	NUM
ejde-942	336	41	n	n	NUM
ejde-942	336	42	/	/	SYM
ejde-942	336	43	a	a	DET
ejde-942	336	44	n	n	NOUN
ejde-942	336	45	/	/	SYM
ejde-942	336	46	a	a	DET
ejde-942	336	47	99.8336	99.8336	NUM
ejde-942	336	48	(	(	PUNCT
ejde-942	336	49	725s	725s	PROPN
ejde-942	336	50	)	)	PUNCT
ejde-942	337	1	ejde-2024/53	ejde-2024/53	ADP
ejde-942	337	2	deep	deep	ADJ
ejde-942	337	3	learning	learning	NOUN
ejde-942	337	4	method	method	NOUN
ejde-942	337	5	for	for	ADP
ejde-942	337	6	finding	find	VERB
ejde-942	337	7	eigenpairs	eigenpair	NOUN
ejde-942	337	8	13	13	NUM
ejde-942	337	9	4.2	4.2	NUM
ejde-942	337	10	.	.	PUNCT
ejde-942	338	1	comparison	comparison	NOUN
ejde-942	338	2	with	with	ADP
ejde-942	338	3	classical	classical	ADJ
ejde-942	338	4	numerical	numerical	ADJ
ejde-942	338	5	methods	method	NOUN
ejde-942	338	6	.	.	PUNCT
ejde-942	339	1	the	the	DET
ejde-942	339	2	following	follow	VERB
ejde-942	339	3	finite	finite	ADJ
ejde-942	339	4	difference	difference	NOUN
ejde-942	339	5	method	method	NOUN
ejde-942	339	6	’s	’s	PART
ejde-942	339	7	discretization	discretization	NOUN
ejde-942	339	8	scheme	scheme	NOUN
ejde-942	339	9	is	be	AUX
ejde-942	339	10	derived	derive	VERB
ejde-942	339	11	from	from	ADP
ejde-942	339	12	reference	reference	NOUN
ejde-942	339	13	[	[	X
ejde-942	339	14	1	1	NUM
ejde-942	339	15	]	]	PUNCT
ejde-942	339	16	.	.	PUNCT
ejde-942	340	1	when	when	SCONJ
ejde-942	340	2	a1	a1	NOUN
ejde-942	340	3	=	=	SYM
ejde-942	340	4	a2	a2	PROPN
ejde-942	340	5	=	=	SYM
ejde-942	340	6	0	0	NUM
ejde-942	340	7	,	,	PUNCT
ejde-942	340	8	we	we	PRON
ejde-942	340	9	have	have	VERB
ejde-942	340	10	dirichlet	dirichlet	PROPN
ejde-942	340	11	boundary	boundary	ADJ
ejde-942	340	12	value	value	NOUN
ejde-942	340	13	condition	condition	NOUN
ejde-942	340	14	.	.	PUNCT
ejde-942	341	1	φ(0	φ(0	ADJ
ejde-942	341	2	)	)	PUNCT
ejde-942	341	3	=	=	SYM
ejde-942	341	4	φ(π	φ(π	NOUN
ejde-942	341	5	)	)	PUNCT
ejde-942	341	6	=	=	SYM
ejde-942	342	1	0	0	X
ejde-942	342	2	.	.	PUNCT
ejde-942	343	1	(	(	PUNCT
ejde-942	343	2	4.3	4.3	NUM
ejde-942	343	3	)	)	PUNCT
ejde-942	343	4	if	if	SCONJ
ejde-942	343	5	,	,	PUNCT
ejde-942	343	6	at	at	ADP
ejde-942	343	7	the	the	DET
ejde-942	343	8	internal	internal	ADJ
ejde-942	343	9	grid	grid	NOUN
ejde-942	343	10	points	point	NOUN
ejde-942	343	11	of	of	ADP
ejde-942	343	12	the	the	DET
ejde-942	343	13	uniform	uniform	NOUN
ejde-942	343	14	grid	grid	NOUN
ejde-942	343	15	g	g	PROPN
ejde-942	343	16	=	=	PUNCT
ejde-942	343	17	{	{	PUNCT
ejde-942	343	18	xi;xi	xi;xi	NOUN
ejde-942	343	19	=	=	SYM
ejde-942	343	20	ih	ih	X
ejde-942	343	21	,	,	PUNCT
ejde-942	343	22	i	i	PRON
ejde-942	343	23	=	=	NOUN
ejde-942	343	24	0	0	NUM
ejde-942	343	25	,	,	PUNCT
ejde-942	343	26	1	1	NUM
ejde-942	343	27	,	,	PUNCT
ejde-942	343	28	.	.	PUNCT
ejde-942	343	29	.	.	PUNCT
ejde-942	343	30	.	.	PUNCT
ejde-942	343	31	,	,	PUNCT
ejde-942	343	32	n	n	CCONJ
ejde-942	343	33	,	,	PUNCT
ejde-942	343	34	n+	n+	ADP
ejde-942	343	35	1	1	NUM
ejde-942	343	36	,	,	PUNCT
ejde-942	343	37	h	h	NOUN
ejde-942	343	38	=	=	SYM
ejde-942	343	39	π/(n+	π/(n+	PROPN
ejde-942	343	40	1	1	NUM
ejde-942	343	41	)	)	PUNCT
ejde-942	343	42	}	}	PUNCT
ejde-942	343	43	,	,	PUNCT
ejde-942	343	44	(	(	PUNCT
ejde-942	343	45	4.4	4.4	NUM
ejde-942	343	46	)	)	PUNCT
ejde-942	343	47	the	the	DET
ejde-942	343	48	operator	operator	NOUN
ejde-942	343	49	lφ	lφ	NOUN
ejde-942	343	50	is	be	AUX
ejde-942	343	51	approximated	approximate	VERB
ejde-942	343	52	using	use	VERB
ejde-942	343	53	central	central	ADJ
ejde-942	343	54	differences	difference	NOUN
ejde-942	343	55	,	,	PUNCT
ejde-942	343	56	with	with	ADP
ejde-942	343	57	l	l	NOUN
ejde-942	343	58	=	=	SYM
ejde-942	343	59			NOUN
ejde-942	343	60	2	2	NUM
ejde-942	343	61	h2	h2	NOUN
ejde-942	343	62	+	+	CCONJ
ejde-942	343	63	ηρ(x1	ηρ(x1	NOUN
ejde-942	343	64	)	)	PUNCT
ejde-942	343	65	−	−	PROPN
ejde-942	343	66	1	1	NUM
ejde-942	343	67	h2	h2	NOUN
ejde-942	343	68	−	−	PROPN
ejde-942	343	69	1	1	NUM
ejde-942	343	70	h2	h2	NOUN
ejde-942	343	71	2	2	NUM
ejde-942	343	72	h2	h2	NOUN
ejde-942	343	73	+	+	CCONJ
ejde-942	343	74	ηρ(x2	ηρ(x2	NOUN
ejde-942	343	75	)	)	PUNCT
ejde-942	343	76	−	−	PROPN
ejde-942	343	77	1	1	NUM
ejde-942	343	78	h2	h2	NOUN
ejde-942	343	79	.	.	PUNCT
ejde-942	343	80	.	.	PUNCT
ejde-942	343	81	.	.	PUNCT
ejde-942	343	82	.	.	PUNCT
ejde-942	343	83	.	.	PUNCT
ejde-942	343	84	.	.	PUNCT
ejde-942	343	85	.	.	PUNCT
ejde-942	343	86	.	.	PUNCT
ejde-942	344	1	.	.	PUNCT
ejde-942	345	1	−	−	NOUN
ejde-942	345	2	1	1	NUM
ejde-942	345	3	h2	h2	NOUN
ejde-942	345	4	2	2	NUM
ejde-942	345	5	h2	h2	NOUN
ejde-942	345	6	+	+	CCONJ
ejde-942	345	7	ηρ(xn−1	ηρ(xn−1	SYM
ejde-942	345	8	)	)	PUNCT
ejde-942	345	9	−	−	PROPN
ejde-942	345	10	1	1	NUM
ejde-942	345	11	h2	h2	NOUN
ejde-942	345	12	−	−	PROPN
ejde-942	345	13	1	1	NUM
ejde-942	345	14	h2	h2	NOUN
ejde-942	345	15	2	2	NUM
ejde-942	345	16	h2	h2	NOUN
ejde-942	345	17	+	+	CCONJ
ejde-942	345	18	ηρ(xn	ηρ(xn	NOUN
ejde-942	345	19	)	)	PUNCT
ejde-942	345	20			NOUN
ejde-942	345	21	.	.	PUNCT
ejde-942	346	1	(	(	PUNCT
ejde-942	346	2	4.5	4.5	NUM
ejde-942	346	3	)	)	PUNCT
ejde-942	346	4	such	such	DET
ejde-942	346	5	a	a	DET
ejde-942	346	6	matrix	matrix	NOUN
ejde-942	346	7	is	be	AUX
ejde-942	346	8	symmetric	symmetric	ADJ
ejde-942	346	9	,	,	PUNCT
ejde-942	346	10	ensuring	ensure	VERB
ejde-942	346	11	the	the	DET
ejde-942	346	12	orthogonality	orthogonality	NOUN
ejde-942	346	13	of	of	ADP
ejde-942	346	14	different	different	ADJ
ejde-942	346	15	eigenfunctions	eigenfunction	NOUN
ejde-942	346	16	.	.	PUNCT
ejde-942	347	1	the	the	DET
ejde-942	347	2	error	error	NOUN
ejde-942	347	3	of	of	ADP
ejde-942	347	4	the	the	DET
ejde-942	347	5	k	k	NOUN
ejde-942	347	6	-	-	PUNCT
ejde-942	347	7	th	th	VERB
ejde-942	347	8	eigenvalue	eigenvalue	NOUN
ejde-942	347	9	λk	λk	NOUN
ejde-942	347	10	is	be	AUX
ejde-942	347	11	about	about	ADP
ejde-942	347	12	o(k4h2	o(k4h2	NOUN
ejde-942	347	13	)	)	PUNCT
ejde-942	347	14	.	.	PUNCT
ejde-942	348	1	however	however	ADV
ejde-942	348	2	,	,	PUNCT
ejde-942	348	3	for	for	ADP
ejde-942	348	4	other	other	ADJ
ejde-942	348	5	boundary	boundary	ADJ
ejde-942	348	6	value	value	NOUN
ejde-942	348	7	conditions	condition	NOUN
ejde-942	348	8	,	,	PUNCT
ejde-942	348	9	especially	especially	ADV
ejde-942	348	10	those	those	PRON
ejde-942	348	11	involving	involve	VERB
ejde-942	348	12	derivatives(a1	derivatives(a1	NOUN
ejde-942	348	13	̸=	̸=	PROPN
ejde-942	348	14	0	0	NUM
ejde-942	348	15	or	or	CCONJ
ejde-942	348	16	a2	a2	PROPN
ejde-942	348	17	̸=	̸=	PROPN
ejde-942	348	18	0	0	NUM
ejde-942	348	19	)	)	PUNCT
ejde-942	348	20	,	,	PUNCT
ejde-942	348	21	loss	loss	NOUN
ejde-942	348	22	of	of	ADP
ejde-942	348	23	symmetry	symmetry	NOUN
ejde-942	348	24	brings	bring	VERB
ejde-942	348	25	additional	additional	ADJ
ejde-942	348	26	difficulties	difficulty	NOUN
ejde-942	348	27	when	when	SCONJ
ejde-942	348	28	using	use	VERB
ejde-942	348	29	classical	classical	ADJ
ejde-942	348	30	numerical	numerical	ADJ
ejde-942	348	31	methods	method	NOUN
ejde-942	348	32	.	.	PUNCT
ejde-942	349	1	when	when	SCONJ
ejde-942	349	2	a1	a1	PROPN
ejde-942	349	3	̸=	̸=	PROPN
ejde-942	349	4	0	0	NUM
ejde-942	349	5	or	or	CCONJ
ejde-942	349	6	a2	a2	PROPN
ejde-942	349	7	̸=	̸=	PROPN
ejde-942	349	8	0	0	NUM
ejde-942	349	9	,	,	PUNCT
ejde-942	349	10	if	if	SCONJ
ejde-942	349	11	we	we	PRON
ejde-942	349	12	use	use	VERB
ejde-942	349	13	finite	finite	ADJ
ejde-942	349	14	difference	difference	NOUN
ejde-942	349	15	method	method	NOUN
ejde-942	349	16	,	,	PUNCT
ejde-942	349	17	we	we	PRON
ejde-942	349	18	found	find	VERB
ejde-942	349	19	that	that	SCONJ
ejde-942	349	20	it	it	PRON
ejde-942	349	21	is	be	AUX
ejde-942	349	22	not	not	PART
ejde-942	349	23	symmetric	symmetric	ADJ
ejde-942	349	24	,	,	PUNCT
ejde-942	349	25	resulting	result	VERB
ejde-942	349	26	in	in	ADP
ejde-942	349	27	the	the	DET
ejde-942	349	28	eigenfunctions	eigenfunction	NOUN
ejde-942	349	29	not	not	PART
ejde-942	349	30	being	be	AUX
ejde-942	349	31	orthogonal	orthogonal	ADJ
ejde-942	349	32	,	,	PUNCT
ejde-942	349	33	while	while	SCONJ
ejde-942	349	34	the	the	DET
ejde-942	349	35	original	original	ADJ
ejde-942	349	36	differential	differential	NOUN
ejde-942	349	37	operator	operator	NOUN
ejde-942	349	38	is	be	AUX
ejde-942	349	39	self	self	NOUN
ejde-942	349	40	-	-	PUNCT
ejde-942	349	41	adjoint	adjoint	NOUN
ejde-942	349	42	,	,	PUNCT
ejde-942	349	43	ensuring	ensure	VERB
ejde-942	349	44	the	the	DET
ejde-942	349	45	orthogonality	orthogonality	NOUN
ejde-942	349	46	of	of	ADP
ejde-942	349	47	the	the	DET
ejde-942	349	48	real	real	ADJ
ejde-942	349	49	eigenfunctions	eigenfunction	NOUN
ejde-942	349	50	.	.	PUNCT
ejde-942	350	1	specifically	specifically	ADV
ejde-942	350	2	,	,	PUNCT
ejde-942	350	3	the	the	DET
ejde-942	350	4	central	central	ADJ
ejde-942	350	5	difference	difference	NOUN
ejde-942	350	6	eigenvalues	eigenvalue	VERB
ejde-942	350	7	λk	λk	ADP
ejde-942	350	8	,	,	PUNCT
ejde-942	350	9	k	k	PROPN
ejde-942	350	10	=	=	SYM
ejde-942	350	11	1	1	NUM
ejde-942	350	12	,	,	PUNCT
ejde-942	350	13	.	.	PUNCT
ejde-942	350	14	.	.	PUNCT
ejde-942	351	1	.	.	PUNCT
ejde-942	352	1	,	,	PUNCT
ejde-942	352	2	n	n	PROPN
ejde-942	352	3	+	+	NOUN
ejde-942	352	4	1	1	NUM
ejde-942	352	5	of	of	ADP
ejde-942	352	6	the	the	DET
ejde-942	352	7	above	above	ADJ
ejde-942	352	8	equation	equation	NOUN
ejde-942	352	9	satisfy	satisfy	VERB
ejde-942	352	10	on	on	ADP
ejde-942	352	11	the	the	DET
ejde-942	352	12	augmented	augment	VERB
ejde-942	352	13	grid	grid	NOUN
ejde-942	352	14	g∗	g∗	NOUN
ejde-942	352	15	=	=	SYM
ejde-942	352	16	{	{	PUNCT
ejde-942	352	17	xj	xj	PROPN
ejde-942	352	18	;	;	PUNCT
ejde-942	352	19	xj	xj	PROPN
ejde-942	352	20	=	=	PROPN
ejde-942	352	21	jh	jh	PROPN
ejde-942	352	22	,	,	PUNCT
ejde-942	352	23	j	j	PROPN
ejde-942	352	24	=	=	SYM
ejde-942	352	25	−1	−1	PROPN
ejde-942	352	26	,	,	PUNCT
ejde-942	352	27	0	0	NUM
ejde-942	352	28	,	,	PUNCT
ejde-942	352	29	1	1	NUM
ejde-942	352	30	,	,	PUNCT
ejde-942	352	31	.	.	PUNCT
ejde-942	352	32	.	.	PUNCT
ejde-942	353	1	.	.	PUNCT
ejde-942	354	1	,	,	PUNCT
ejde-942	354	2	n	n	CCONJ
ejde-942	354	3	,	,	PUNCT
ejde-942	354	4	n+	n+	ADP
ejde-942	354	5	1	1	NUM
ejde-942	354	6	,	,	PUNCT
ejde-942	354	7	h	h	NOUN
ejde-942	355	1	=	=	SYM
ejde-942	355	2	π	π	PROPN
ejde-942	355	3	/	/	SYM
ejde-942	355	4	n	n	CCONJ
ejde-942	355	5	}	}	PUNCT
ejde-942	355	6	,	,	PUNCT
ejde-942	355	7	(	(	PUNCT
ejde-942	355	8	4.6	4.6	NUM
ejde-942	355	9	)	)	PUNCT
ejde-942	355	10	the	the	DET
ejde-942	355	11	following	follow	VERB
ejde-942	355	12	eigenvalue	eigenvalue	PROPN
ejde-942	355	13	problem	problem	NOUN
ejde-942	355	14	of	of	ADP
ejde-942	355	15	matrix	matrix	NOUN
ejde-942	355	16	l	l	NOUN
ejde-942	355	17	=	=	SYM
ejde-942	355	18			NOUN
ejde-942	355	19	2	2	NUM
ejde-942	355	20	h2	h2	NOUN
ejde-942	355	21	+	+	CCONJ
ejde-942	355	22	ηρ(x0	ηρ(x0	NOUN
ejde-942	355	23	)	)	PUNCT
ejde-942	355	24	−	−	PROPN
ejde-942	355	25	α	α	PROPN
ejde-942	355	26	h2	h2	NOUN
ejde-942	355	27	−	−	PROPN
ejde-942	355	28	1	1	NUM
ejde-942	355	29	h2	h2	NOUN
ejde-942	355	30	2	2	NUM
ejde-942	355	31	h2	h2	NOUN
ejde-942	355	32	+	+	CCONJ
ejde-942	355	33	ηρ(x1	ηρ(x1	NOUN
ejde-942	355	34	)	)	PUNCT
ejde-942	355	35	−	−	PROPN
ejde-942	355	36	1	1	NUM
ejde-942	355	37	h2	h2	NOUN
ejde-942	355	38	.	.	PUNCT
ejde-942	355	39	.	.	PUNCT
ejde-942	355	40	.	.	PUNCT
ejde-942	355	41	.	.	PUNCT
ejde-942	355	42	.	.	PUNCT
ejde-942	355	43	.	.	PUNCT
ejde-942	355	44	.	.	PUNCT
ejde-942	355	45	.	.	PUNCT
ejde-942	355	46	.	.	PUNCT
ejde-942	356	1	−	−	NOUN
ejde-942	356	2	1	1	NUM
ejde-942	356	3	h2	h2	NOUN
ejde-942	356	4	2	2	NUM
ejde-942	356	5	h2	h2	NOUN
ejde-942	356	6	+	+	CCONJ
ejde-942	356	7	ηρ(xn−1	ηρ(xn−1	SYM
ejde-942	356	8	)	)	PUNCT
ejde-942	356	9	−	−	PROPN
ejde-942	356	10	1	1	NUM
ejde-942	356	11	h2	h2	NOUN
ejde-942	356	12	−	−	PROPN
ejde-942	356	13	β	β	SYM
ejde-942	356	14	h2	h2	PROPN
ejde-942	356	15	2	2	NUM
ejde-942	356	16	h2	h2	NOUN
ejde-942	356	17	+	+	CCONJ
ejde-942	356	18	ηρ(xn	ηρ(xn	NOUN
ejde-942	356	19	)	)	PUNCT
ejde-942	356	20			NOUN
ejde-942	356	21	(	(	PUNCT
ejde-942	356	22	4.7	4.7	NUM
ejde-942	356	23	)	)	PUNCT
ejde-942	356	24	with	with	ADP
ejde-942	356	25	α	α	NOUN
ejde-942	356	26	=	=	SYM
ejde-942	356	27	2a1	2a1	NUM
ejde-942	356	28	a1	a1	NOUN
ejde-942	356	29	+	+	CCONJ
ejde-942	356	30	h	h	NOUN
ejde-942	356	31	(	(	PUNCT
ejde-942	356	32	b1	b1	NOUN
ejde-942	356	33	+	+	CCONJ
ejde-942	356	34	a1	a1	NOUN
ejde-942	356	35	2	2	NUM
ejde-942	356	36	ρ′(0	ρ′(0	NOUN
ejde-942	356	37	)	)	PUNCT
ejde-942	356	38	ρ(0	ρ(0	PROPN
ejde-942	356	39	)	)	PUNCT
ejde-942	356	40	)	)	PUNCT
ejde-942	356	41	and	and	CCONJ
ejde-942	356	42	β	β	X
ejde-942	356	43	=	=	SYM
ejde-942	356	44	2a2	2a2	NUM
ejde-942	356	45	a2	a2	PROPN
ejde-942	356	46	+	+	CCONJ
ejde-942	356	47	h	h	PROPN
ejde-942	356	48	(	(	PUNCT
ejde-942	356	49	b2	b2	NOUN
ejde-942	356	50	−	−	PROPN
ejde-942	356	51	a2	a2	PROPN
ejde-942	356	52	2	2	NUM
ejde-942	356	53	ρ′(π	ρ′(π	NOUN
ejde-942	356	54	)	)	PUNCT
ejde-942	356	55	ρ(π	ρ(π	PROPN
ejde-942	356	56	)	)	PUNCT
ejde-942	356	57	)	)	PUNCT
ejde-942	356	58	.	.	PUNCT
ejde-942	357	1	obviously	obviously	ADV
ejde-942	357	2	,	,	PUNCT
ejde-942	357	3	it	it	PRON
ejde-942	357	4	is	be	AUX
ejde-942	357	5	not	not	PART
ejde-942	357	6	a	a	DET
ejde-942	357	7	diagonal	diagonal	ADJ
ejde-942	357	8	matrix	matrix	NOUN
ejde-942	357	9	.	.	PUNCT
ejde-942	358	1	although	although	SCONJ
ejde-942	358	2	the	the	DET
ejde-942	358	3	eigenvalues	eigenvalue	NOUN
ejde-942	358	4	we	we	PRON
ejde-942	358	5	calculated	calculate	VERB
ejde-942	358	6	are	be	AUX
ejde-942	358	7	correct	correct	ADJ
ejde-942	358	8	,	,	PUNCT
ejde-942	358	9	the	the	DET
ejde-942	358	10	corresponding	corresponding	ADJ
ejde-942	358	11	eigenfunctions	eigenfunction	NOUN
ejde-942	358	12	are	be	AUX
ejde-942	358	13	not	not	PART
ejde-942	358	14	orthogonal	orthogonal	ADJ
ejde-942	358	15	,	,	PUNCT
ejde-942	358	16	which	which	PRON
ejde-942	358	17	limits	limit	VERB
ejde-942	358	18	the	the	DET
ejde-942	358	19	application	application	NOUN
ejde-942	358	20	of	of	ADP
ejde-942	358	21	such	such	DET
ejde-942	358	22	a	a	DET
ejde-942	358	23	method	method	NOUN
ejde-942	358	24	.	.	PUNCT
ejde-942	359	1	despite	despite	SCONJ
ejde-942	359	2	the	the	DET
ejde-942	359	3	fact	fact	NOUN
ejde-942	359	4	that	that	SCONJ
ejde-942	359	5	the	the	DET
ejde-942	359	6	study	study	NOUN
ejde-942	359	7	of	of	ADP
ejde-942	359	8	the	the	DET
ejde-942	359	9	dirichlet	dirichlet	PROPN
ejde-942	359	10	eigenvalue	eigenvalue	PROPN
ejde-942	359	11	problem	problem	NOUN
ejde-942	359	12	using	use	VERB
ejde-942	359	13	deep	deep	ADJ
ejde-942	359	14	learning	learning	NOUN
ejde-942	359	15	methods	method	NOUN
ejde-942	359	16	is	be	AUX
ejde-942	359	17	a	a	DET
ejde-942	359	18	hot	hot	ADJ
ejde-942	359	19	topic	topic	NOUN
ejde-942	359	20	,	,	PUNCT
ejde-942	359	21	to	to	ADP
ejde-942	359	22	our	our	PRON
ejde-942	359	23	knowledge	knowledge	NOUN
ejde-942	359	24	,	,	PUNCT
ejde-942	359	25	there	there	PRON
ejde-942	359	26	is	be	VERB
ejde-942	359	27	no	no	DET
ejde-942	359	28	work	work	NOUN
ejde-942	359	29	on	on	ADP
ejde-942	359	30	the	the	DET
ejde-942	359	31	boundary	boundary	ADJ
ejde-942	359	32	value	value	NOUN
ejde-942	359	33	condition	condition	NOUN
ejde-942	359	34	involving	involve	VERB
ejde-942	359	35	derivatives	derivative	NOUN
ejde-942	359	36	using	use	VERB
ejde-942	359	37	the	the	DET
ejde-942	359	38	deep	deep	ADJ
ejde-942	359	39	learning	learning	NOUN
ejde-942	359	40	method	method	NOUN
ejde-942	359	41	.	.	PUNCT
ejde-942	360	1	indeed	indeed	ADV
ejde-942	360	2	,	,	PUNCT
ejde-942	360	3	the	the	DET
ejde-942	360	4	method	method	NOUN
ejde-942	360	5	we	we	PRON
ejde-942	360	6	present	present	VERB
ejde-942	360	7	in	in	ADP
ejde-942	360	8	this	this	DET
ejde-942	360	9	paper	paper	NOUN
ejde-942	360	10	can	can	AUX
ejde-942	360	11	easily	easily	ADV
ejde-942	360	12	tackle	tackle	VERB
ejde-942	360	13	boundary	boundary	ADJ
ejde-942	360	14	value	value	NOUN
ejde-942	360	15	conditions	condition	NOUN
ejde-942	360	16	with	with	ADP
ejde-942	360	17	derivatives	derivative	NOUN
ejde-942	360	18	,	,	PUNCT
ejde-942	360	19	resulting	result	VERB
ejde-942	360	20	in	in	ADP
ejde-942	360	21	orthogonal	orthogonal	ADJ
ejde-942	360	22	eigenfunctions	eigenfunction	NOUN
ejde-942	360	23	.	.	PUNCT
ejde-942	361	1	this	this	PRON
ejde-942	361	2	is	be	AUX
ejde-942	361	3	a	a	DET
ejde-942	361	4	very	very	ADV
ejde-942	361	5	important	important	ADJ
ejde-942	361	6	advantage	advantage	NOUN
ejde-942	361	7	of	of	ADP
ejde-942	361	8	deep	deep	ADJ
ejde-942	361	9	learning	learning	NOUN
ejde-942	361	10	methods	method	NOUN
ejde-942	361	11	that	that	PRON
ejde-942	361	12	has	have	AUX
ejde-942	361	13	not	not	PART
ejde-942	361	14	yet	yet	ADV
ejde-942	361	15	been	be	AUX
ejde-942	361	16	noticed	notice	VERB
ejde-942	361	17	.	.	PUNCT
ejde-942	362	1	additionally	additionally	ADV
ejde-942	362	2	,	,	PUNCT
ejde-942	362	3	we	we	PRON
ejde-942	362	4	do	do	AUX
ejde-942	362	5	n’t	not	PART
ejde-942	362	6	require	require	VERB
ejde-942	362	7	ρ(x	ρ(x	NOUN
ejde-942	362	8	)	)	PUNCT
ejde-942	362	9	to	to	PART
ejde-942	362	10	have	have	VERB
ejde-942	362	11	a	a	DET
ejde-942	362	12	second	second	ADJ
ejde-942	362	13	-	-	PUNCT
ejde-942	362	14	order	order	NOUN
ejde-942	362	15	derivative	derivative	NOUN
ejde-942	362	16	,	,	PUNCT
ejde-942	362	17	which	which	PRON
ejde-942	362	18	is	be	AUX
ejde-942	362	19	an	an	DET
ejde-942	362	20	assumption	assumption	NOUN
ejde-942	362	21	required	require	VERB
ejde-942	362	22	by	by	ADP
ejde-942	362	23	the	the	DET
ejde-942	362	24	above	above	ADJ
ejde-942	362	25	method	method	NOUN
ejde-942	362	26	.	.	PUNCT
ejde-942	363	1	unlike	unlike	ADP
ejde-942	363	2	classical	classical	ADJ
ejde-942	363	3	numerical	numerical	ADJ
ejde-942	363	4	methods	method	NOUN
ejde-942	363	5	,	,	PUNCT
ejde-942	363	6	our	our	PRON
ejde-942	363	7	neural	neural	ADJ
ejde-942	363	8	network	network	NOUN
ejde-942	363	9	solution	solution	NOUN
ejde-942	363	10	provides	provide	VERB
ejde-942	363	11	an	an	DET
ejde-942	363	12	approximate	approximate	ADJ
ejde-942	363	13	analytical	analytical	ADJ
ejde-942	363	14	solution	solution	NOUN
ejde-942	363	15	,	,	PUNCT
ejde-942	363	16	in	in	ADP
ejde-942	363	17	a	a	DET
ejde-942	363	18	certain	certain	ADJ
ejde-942	363	19	sense	sense	NOUN
ejde-942	363	20	,	,	PUNCT
ejde-942	363	21	which	which	PRON
ejde-942	363	22	allows	allow	VERB
ejde-942	363	23	us	we	PRON
ejde-942	363	24	to	to	PART
ejde-942	363	25	obtain	obtain	VERB
ejde-942	363	26	derivatives	derivative	NOUN
ejde-942	363	27	using	use	VERB
ejde-942	363	28	automatic	automatic	ADJ
ejde-942	363	29	differentiation	differentiation	NOUN
ejde-942	363	30	.	.	PUNCT
ejde-942	364	1	14	14	NUM
ejde-942	364	2	s.	s.	PROPN
ejde-942	364	3	zhang	zhang	PROPN
ejde-942	364	4	,	,	PUNCT
ejde-942	364	5	j.	j.	PROPN
ejde-942	364	6	zu	zu	PROPN
ejde-942	364	7	,	,	PUNCT
ejde-942	364	8	j.	j.	PROPN
ejde-942	364	9	zhang	zhang	PROPN
ejde-942	364	10	ejde-2024/53	ejde-2024/53	PROPN
ejde-942	364	11	5	5	NUM
ejde-942	364	12	.	.	PUNCT
ejde-942	364	13	numerical	numerical	ADJ
ejde-942	364	14	experiments	experiment	NOUN
ejde-942	364	15	in	in	ADP
ejde-942	364	16	this	this	DET
ejde-942	364	17	section	section	NOUN
ejde-942	364	18	,	,	PUNCT
ejde-942	364	19	we	we	PRON
ejde-942	364	20	will	will	AUX
ejde-942	364	21	utilize	utilize	VERB
ejde-942	364	22	the	the	DET
ejde-942	364	23	designed	design	VERB
ejde-942	364	24	neural	neural	ADJ
ejde-942	364	25	network	network	NOUN
ejde-942	364	26	algorithm	algorithm	NOUN
ejde-942	364	27	to	to	PART
ejde-942	364	28	identify	identify	VERB
ejde-942	364	29	the	the	DET
ejde-942	364	30	smallest	small	ADJ
ejde-942	364	31	eigenpairs	eigenpair	NOUN
ejde-942	364	32	under	under	ADP
ejde-942	364	33	various	various	ADJ
ejde-942	364	34	boundary	boundary	ADJ
ejde-942	364	35	value	value	NOUN
ejde-942	364	36	conditions	condition	NOUN
ejde-942	364	37	.	.	PUNCT
ejde-942	365	1	the	the	DET
ejde-942	365	2	numerical	numerical	ADJ
ejde-942	365	3	results	result	NOUN
ejde-942	365	4	obtained	obtain	VERB
ejde-942	365	5	from	from	ADP
ejde-942	365	6	the	the	DET
ejde-942	365	7	neural	neural	ADJ
ejde-942	365	8	network	network	NOUN
ejde-942	365	9	algorithm	algorithm	NOUN
ejde-942	365	10	will	will	AUX
ejde-942	365	11	be	be	AUX
ejde-942	365	12	compared	compare	VERB
ejde-942	365	13	to	to	ADP
ejde-942	365	14	those	those	PRON
ejde-942	365	15	from	from	ADP
ejde-942	365	16	the	the	DET
ejde-942	365	17	classical	classical	ADJ
ejde-942	365	18	numerical	numerical	ADJ
ejde-942	365	19	method	method	NOUN
ejde-942	365	20	,	,	PUNCT
ejde-942	365	21	demonstrating	demonstrate	VERB
ejde-942	365	22	the	the	DET
ejde-942	365	23	efficiency	efficiency	NOUN
ejde-942	365	24	of	of	ADP
ejde-942	365	25	our	our	PRON
ejde-942	365	26	approach	approach	NOUN
ejde-942	365	27	.	.	PUNCT
ejde-942	366	1	moreover	moreover	ADV
ejde-942	366	2	,	,	PUNCT
ejde-942	366	3	we	we	PRON
ejde-942	366	4	illustrate	illustrate	VERB
ejde-942	366	5	that	that	SCONJ
ejde-942	366	6	our	our	PRON
ejde-942	366	7	proposed	propose	VERB
ejde-942	366	8	method	method	NOUN
ejde-942	366	9	can	can	AUX
ejde-942	366	10	be	be	AUX
ejde-942	366	11	seamlessly	seamlessly	ADV
ejde-942	366	12	extended	extend	VERB
ejde-942	366	13	to	to	PART
ejde-942	366	14	handle	handle	VERB
ejde-942	366	15	two	two	NUM
ejde-942	366	16	-	-	PUNCT
ejde-942	366	17	dimensional	dimensional	ADJ
ejde-942	366	18	cases	case	NOUN
ejde-942	366	19	and	and	CCONJ
ejde-942	366	20	periodic	periodic	ADJ
ejde-942	366	21	scenarios	scenario	NOUN
ejde-942	366	22	.	.	PUNCT
ejde-942	367	1	example	example	NOUN
ejde-942	367	2	5.1	5.1	NUM
ejde-942	367	3	.	.	PUNCT
ejde-942	368	1	the	the	DET
ejde-942	368	2	sturm	sturm	NOUN
ejde-942	368	3	-	-	PUNCT
ejde-942	368	4	liouville	liouville	NOUN
ejde-942	368	5	problem	problem	NOUN
ejde-942	368	6	with	with	ADP
ejde-942	368	7	neumann	neumann	PROPN
ejde-942	368	8	boundary	boundary	ADJ
ejde-942	368	9	value	value	NOUN
ejde-942	368	10	conditions	condition	NOUN
ejde-942	368	11	:	:	PUNCT
ejde-942	368	12	−(ρ(x)φ′(x))′	−(ρ(x)φ′(x))′	PROPN
ejde-942	368	13	=	=	SYM
ejde-942	368	14	λρ(x)φ(x	λρ(x)φ(x	PROPN
ejde-942	368	15	)	)	PUNCT
ejde-942	368	16	,	,	PUNCT
ejde-942	369	1	x	x	PUNCT
ejde-942	369	2	∈	∈	PROPN
ejde-942	369	3	(	(	PUNCT
ejde-942	369	4	0	0	NUM
ejde-942	369	5	,	,	PUNCT
ejde-942	369	6	π	π	NOUN
ejde-942	369	7	)	)	PUNCT
ejde-942	369	8	,	,	PUNCT
ejde-942	369	9	φ′(0	φ′(0	X
ejde-942	369	10	)	)	PUNCT
ejde-942	369	11	=	=	SYM
ejde-942	369	12	φ′(π	φ′(π	PROPN
ejde-942	369	13	)	)	PUNCT
ejde-942	369	14	=	=	SYM
ejde-942	369	15	0	0	NUM
ejde-942	369	16	,	,	PUNCT
ejde-942	369	17	(	(	PUNCT
ejde-942	369	18	5.1	5.1	NUM
ejde-942	369	19	)	)	PUNCT
ejde-942	369	20	where	where	SCONJ
ejde-942	369	21	ρ(x	ρ(x	NOUN
ejde-942	369	22	)	)	PUNCT
ejde-942	369	23	=	=	SYM
ejde-942	370	1	1	1	NUM
ejde-942	370	2	+	+	NUM
ejde-942	370	3	0.3	0.3	NUM
ejde-942	370	4	sinx	sinx	NOUN
ejde-942	370	5	.	.	PUNCT
ejde-942	371	1	table	table	NOUN
ejde-942	371	2	5.1	5.1	NUM
ejde-942	371	3	shows	show	VERB
ejde-942	371	4	the	the	DET
ejde-942	371	5	eigenvalues	eigenvalue	NOUN
ejde-942	371	6	of	of	ADP
ejde-942	371	7	the	the	DET
ejde-942	371	8	sturm	sturm	NOUN
ejde-942	371	9	-	-	PUNCT
ejde-942	371	10	liouville	liouville	NOUN
ejde-942	371	11	problem	problem	NOUN
ejde-942	371	12	(	(	PUNCT
ejde-942	371	13	5.1	5.1	NUM
ejde-942	371	14	)	)	PUNCT
ejde-942	371	15	.	.	PUNCT
ejde-942	372	1	λ̃n	λ̃n	PROPN
ejde-942	372	2	represents	represent	VERB
ejde-942	372	3	the	the	DET
ejde-942	372	4	numerical	numerical	ADJ
ejde-942	372	5	results	result	NOUN
ejde-942	372	6	obtained	obtain	VERB
ejde-942	372	7	via	via	ADP
ejde-942	372	8	the	the	DET
ejde-942	372	9	finite	finite	ADJ
ejde-942	372	10	difference	difference	NOUN
ejde-942	372	11	method	method	NOUN
ejde-942	372	12	(	(	PUNCT
ejde-942	372	13	4.5	4.5	NUM
ejde-942	372	14	)	)	PUNCT
ejde-942	372	15	with	with	ADP
ejde-942	372	16	h	h	NOUN
ejde-942	372	17	=	=	SYM
ejde-942	372	18	π	π	PROPN
ejde-942	372	19	1000	1000	NUM
ejde-942	372	20	.	.	PUNCT
ejde-942	373	1	λ̂n	λ̂n	PUNCT
ejde-942	373	2	represents	represent	VERB
ejde-942	373	3	the	the	DET
ejde-942	373	4	numerical	numerical	ADJ
ejde-942	373	5	results	result	NOUN
ejde-942	373	6	obtained	obtain	VERB
ejde-942	373	7	from	from	ADP
ejde-942	373	8	the	the	DET
ejde-942	373	9	deep	deep	ADJ
ejde-942	373	10	learning	learning	NOUN
ejde-942	373	11	algorithm	algorithm	NOUN
ejde-942	373	12	.	.	PUNCT
ejde-942	374	1	we	we	PRON
ejde-942	374	2	get	get	VERB
ejde-942	374	3	the	the	DET
ejde-942	374	4	smallest	small	ADJ
ejde-942	374	5	eigenvalues	eigenvalue	NOUN
ejde-942	374	6	and	and	CCONJ
ejde-942	374	7	eigenfunctions	eigenfunction	NOUN
ejde-942	374	8	simultaneously	simultaneously	ADV
ejde-942	374	9	,	,	PUNCT
ejde-942	374	10	see	see	VERB
ejde-942	374	11	figure	figure	NOUN
ejde-942	374	12	5.1	5.1	NUM
ejde-942	374	13	.	.	PUNCT
ejde-942	374	14	table	table	NOUN
ejde-942	374	15	5.1	5.1	NUM
ejde-942	374	16	.	.	PUNCT
ejde-942	375	1	eigenvalues	eigenvalue	NOUN
ejde-942	375	2	of	of	ADP
ejde-942	375	3	sturm	sturm	NOUN
ejde-942	375	4	-	-	PUNCT
ejde-942	375	5	liouville	liouville	NOUN
ejde-942	375	6	problem	problem	NOUN
ejde-942	375	7	(	(	PUNCT
ejde-942	375	8	5.1	5.1	NUM
ejde-942	375	9	)	)	PUNCT
ejde-942	375	10	λn	λn	NOUN
ejde-942	375	11	λ1	λ1	PROPN
ejde-942	375	12	λ2	λ2	NOUN
ejde-942	376	1	λ3	λ3	PROPN
ejde-942	376	2	λ4	λ4	PROPN
ejde-942	376	3	λ5	λ5	VERB
ejde-942	376	4	λ6	λ6	PROPN
ejde-942	376	5	λ7	λ7	PROPN
ejde-942	376	6	λ8	λ8	PROPN
ejde-942	376	7	λ9	λ9	ADJ
ejde-942	376	8	λ10	λ10	NOUN
ejde-942	376	9	λ̃n	λ̃n	X
ejde-942	376	10	7.56	7.56	NUM
ejde-942	376	11	·	·	PUNCT
ejde-942	376	12	10−6	10−6	NUM
ejde-942	376	13	1.1129	1.1129	NUM
ejde-942	376	14	4.1098	4.1098	NUM
ejde-942	376	15	9.1099	9.1099	NUM
ejde-942	376	16	16.1119	16.1119	NUM
ejde-942	376	17	25.1151	25.1151	NUM
ejde-942	376	18	36.1193	36.1193	NUM
ejde-942	376	19	49.1243	49.1243	NUM
ejde-942	376	20	64.1302	64.1302	NUM
ejde-942	376	21	81.1368	81.1368	NUM
ejde-942	376	22	λ̂n	λ̂n	PRON
ejde-942	376	23	6.62	6.62	NUM
ejde-942	376	24	·	·	PUNCT
ejde-942	376	25	10−6	10−6	NUM
ejde-942	376	26	1.1126	1.1126	NUM
ejde-942	376	27	4.1082	4.1082	NUM
ejde-942	376	28	9.1077	9.1077	NUM
ejde-942	376	29	16.1057	16.1057	NUM
ejde-942	376	30	25.1079	25.1079	NUM
ejde-942	376	31	36.0993	36.0993	NUM
ejde-942	376	32	49.1056	49.1056	NUM
ejde-942	376	33	64.0550	64.0550	NUM
ejde-942	376	34	81.0975	81.0975	NUM
ejde-942	376	35	figure	figure	NOUN
ejde-942	376	36	5.1	5.1	NUM
ejde-942	376	37	.	.	PUNCT
ejde-942	377	1	eigenfunctions	eigenfunction	NOUN
ejde-942	377	2	of	of	ADP
ejde-942	377	3	the	the	DET
ejde-942	377	4	sturm	sturm	NOUN
ejde-942	377	5	-	-	PUNCT
ejde-942	377	6	liouville	liouville	NOUN
ejde-942	377	7	problem	problem	NOUN
ejde-942	377	8	(	(	PUNCT
ejde-942	377	9	5.1	5.1	NUM
ejde-942	377	10	)	)	PUNCT
ejde-942	377	11	example	example	NOUN
ejde-942	377	12	5.1	5.1	NUM
ejde-942	377	13	shows	show	VERB
ejde-942	377	14	that	that	SCONJ
ejde-942	377	15	our	our	PRON
ejde-942	377	16	neural	neural	ADJ
ejde-942	377	17	network	network	NOUN
ejde-942	377	18	method	method	NOUN
ejde-942	377	19	is	be	AUX
ejde-942	377	20	well	well	ADV
ejde-942	377	21	adapted	adapt	VERB
ejde-942	377	22	to	to	ADP
ejde-942	377	23	the	the	DET
ejde-942	377	24	inhomogeneous	inhomogeneous	ADJ
ejde-942	377	25	media	medium	NOUN
ejde-942	377	26	case	case	NOUN
ejde-942	377	27	,	,	PUNCT
ejde-942	377	28	especially	especially	ADV
ejde-942	377	29	to	to	ADP
ejde-942	377	30	the	the	DET
ejde-942	377	31	derivative	derivative	ADJ
ejde-942	377	32	boundary	boundary	ADJ
ejde-942	377	33	value	value	NOUN
ejde-942	377	34	case	case	NOUN
ejde-942	377	35	.	.	PUNCT
ejde-942	378	1	the	the	DET
ejde-942	378	2	resulting	result	VERB
ejde-942	378	3	eigenfunction	eigenfunction	NOUN
ejde-942	378	4	is	be	AUX
ejde-942	378	5	orthogonal	orthogonal	ADJ
ejde-942	378	6	and	and	CCONJ
ejde-942	378	7	has	have	VERB
ejde-942	378	8	broad	broad	ADJ
ejde-942	378	9	prospects	prospect	NOUN
ejde-942	378	10	for	for	ADP
ejde-942	378	11	use	use	NOUN
ejde-942	378	12	in	in	ADP
ejde-942	378	13	other	other	ADJ
ejde-942	378	14	mathematical	mathematical	ADJ
ejde-942	378	15	problems	problem	NOUN
ejde-942	378	16	.	.	PUNCT
ejde-942	379	1	example	example	NOUN
ejde-942	379	2	5.2	5.2	NUM
ejde-942	379	3	.	.	PUNCT
ejde-942	380	1	consider	consider	VERB
ejde-942	380	2	the	the	DET
ejde-942	380	3	periodic	periodic	ADJ
ejde-942	380	4	eigenpair	eigenpair	NOUN
ejde-942	380	5	problems	problem	NOUN
ejde-942	380	6	:	:	PUNCT
ejde-942	380	7	−φ′′(x	−φ′′(x	X
ejde-942	380	8	)	)	PUNCT
ejde-942	380	9	=	=	SYM
ejde-942	380	10	λφ(x	λφ(x	PROPN
ejde-942	380	11	)	)	PUNCT
ejde-942	380	12	,	,	PUNCT
ejde-942	380	13	x	x	PUNCT
ejde-942	380	14	∈	∈	PROPN
ejde-942	380	15	(	(	PUNCT
ejde-942	380	16	0	0	NUM
ejde-942	380	17	,	,	PUNCT
ejde-942	380	18	π	π	NOUN
ejde-942	380	19	)	)	PUNCT
ejde-942	380	20	,	,	PUNCT
ejde-942	380	21	φ(0	φ(0	ADJ
ejde-942	380	22	)	)	PUNCT
ejde-942	380	23	=	=	SYM
ejde-942	380	24	φ(π	φ(π	PROPN
ejde-942	380	25	)	)	PUNCT
ejde-942	380	26	,	,	PUNCT
ejde-942	380	27	φ′(0	φ′(0	X
ejde-942	380	28	)	)	PUNCT
ejde-942	380	29	=	=	SYM
ejde-942	380	30	φ′(π	φ′(π	NOUN
ejde-942	380	31	)	)	PUNCT
ejde-942	380	32	.	.	PUNCT
ejde-942	381	1	(	(	PUNCT
ejde-942	381	2	5.2	5.2	NUM
ejde-942	381	3	)	)	PUNCT
ejde-942	381	4	ejde-2024/53	ejde-2024/53	VERB
ejde-942	381	5	deep	deep	ADJ
ejde-942	381	6	learning	learning	NOUN
ejde-942	381	7	method	method	NOUN
ejde-942	381	8	for	for	ADP
ejde-942	381	9	finding	find	VERB
ejde-942	381	10	eigenpairs	eigenpair	NOUN
ejde-942	381	11	15	15	NUM
ejde-942	381	12	periodic	periodic	ADJ
ejde-942	381	13	phenomena	phenomenon	NOUN
ejde-942	381	14	are	be	AUX
ejde-942	381	15	ubiquitous	ubiquitous	ADJ
ejde-942	381	16	in	in	ADP
ejde-942	381	17	physics	physics	NOUN
ejde-942	381	18	.	.	PUNCT
ejde-942	382	1	regarding	regard	VERB
ejde-942	382	2	the	the	DET
ejde-942	382	3	periodic	periodic	ADJ
ejde-942	382	4	eigenvalue	eigenvalue	NOUN
ejde-942	382	5	problem	problem	NOUN
ejde-942	382	6	,	,	PUNCT
ejde-942	382	7	where	where	SCONJ
ejde-942	382	8	each	each	DET
ejde-942	382	9	eigenvalue	eigenvalue	NOUN
ejde-942	382	10	corresponds	correspond	VERB
ejde-942	382	11	to	to	ADP
ejde-942	382	12	a	a	DET
ejde-942	382	13	pair	pair	NOUN
ejde-942	382	14	of	of	ADP
ejde-942	382	15	mutually	mutually	ADV
ejde-942	382	16	orthogonal	orthogonal	ADJ
ejde-942	382	17	eigenvectors	eigenvector	NOUN
ejde-942	382	18	,	,	PUNCT
ejde-942	382	19	to	to	ADP
ejde-942	382	20	our	our	PRON
ejde-942	382	21	knowledge	knowledge	NOUN
ejde-942	382	22	,	,	PUNCT
ejde-942	382	23	no	no	DET
ejde-942	382	24	scholar	scholar	NOUN
ejde-942	382	25	has	have	AUX
ejde-942	382	26	yet	yet	ADV
ejde-942	382	27	studied	study	VERB
ejde-942	382	28	how	how	SCONJ
ejde-942	382	29	to	to	PART
ejde-942	382	30	use	use	VERB
ejde-942	382	31	deep	deep	ADJ
ejde-942	382	32	learning	learning	NOUN
ejde-942	382	33	methods	method	NOUN
ejde-942	382	34	to	to	PART
ejde-942	382	35	find	find	VERB
ejde-942	382	36	multiple	multiple	ADJ
ejde-942	382	37	smallest	small	ADJ
ejde-942	382	38	eigenpairs	eigenpair	NOUN
ejde-942	382	39	for	for	ADP
ejde-942	382	40	such	such	ADJ
ejde-942	382	41	problems	problem	NOUN
ejde-942	382	42	.	.	PUNCT
ejde-942	383	1	in	in	ADP
ejde-942	383	2	table	table	NOUN
ejde-942	383	3	5.2	5.2	NUM
ejde-942	383	4	,	,	PUNCT
ejde-942	383	5	λ̂n	λ̂n	PUNCT
ejde-942	383	6	represents	represent	VERB
ejde-942	383	7	the	the	DET
ejde-942	383	8	numerical	numerical	ADJ
ejde-942	383	9	results	result	NOUN
ejde-942	383	10	obtained	obtain	VERB
ejde-942	383	11	from	from	ADP
ejde-942	383	12	the	the	DET
ejde-942	383	13	deep	deep	ADJ
ejde-942	383	14	learning	learning	NOUN
ejde-942	383	15	algorithm	algorithm	NOUN
ejde-942	383	16	,	,	PUNCT
ejde-942	383	17	and	and	CCONJ
ejde-942	383	18	λn	λn	PROPN
ejde-942	383	19	represents	represent	VERB
ejde-942	383	20	the	the	DET
ejde-942	383	21	exact	exact	ADJ
ejde-942	383	22	value	value	NOUN
ejde-942	383	23	of	of	ADP
ejde-942	383	24	the	the	DET
ejde-942	383	25	eigenvalue	eigenvalue	NOUN
ejde-942	383	26	.	.	PUNCT
ejde-942	384	1	it	it	PRON
ejde-942	384	2	shows	show	VERB
ejde-942	384	3	that	that	SCONJ
ejde-942	384	4	our	our	PRON
ejde-942	384	5	algorithm	algorithm	NOUN
ejde-942	384	6	can	can	AUX
ejde-942	384	7	be	be	AUX
ejde-942	384	8	easily	easily	ADV
ejde-942	384	9	applied	apply	VERB
ejde-942	384	10	to	to	ADP
ejde-942	384	11	periodic	periodic	ADJ
ejde-942	384	12	boundary	boundary	ADJ
ejde-942	384	13	value	value	NOUN
ejde-942	384	14	cases	case	NOUN
ejde-942	384	15	with	with	ADP
ejde-942	384	16	enough	enough	ADJ
ejde-942	384	17	accuracy	accuracy	NOUN
ejde-942	384	18	.	.	PUNCT
ejde-942	385	1	table	table	NOUN
ejde-942	385	2	5.2	5.2	NUM
ejde-942	385	3	.	.	PUNCT
ejde-942	386	1	eigenvalues	eigenvalue	NOUN
ejde-942	386	2	of	of	ADP
ejde-942	386	3	periodic	periodic	ADJ
ejde-942	386	4	boundary	boundary	ADJ
ejde-942	386	5	value	value	NOUN
ejde-942	386	6	problems	problem	NOUN
ejde-942	386	7	(	(	PUNCT
ejde-942	386	8	5.2	5.2	NUM
ejde-942	386	9	)	)	PUNCT
ejde-942	386	10	λn	λn	NOUN
ejde-942	386	11	λ1	λ1	PROPN
ejde-942	386	12	λ2	λ2	NOUN
ejde-942	387	1	λ3	λ3	PROPN
ejde-942	387	2	λ4	λ4	PROPN
ejde-942	387	3	λ5	λ5	VERB
ejde-942	387	4	λ6	λ6	PROPN
ejde-942	387	5	λ7	λ7	PROPN
ejde-942	387	6	λ8	λ8	PROPN
ejde-942	387	7	λ9	λ9	ADJ
ejde-942	387	8	λ10	λ10	NOUN
ejde-942	387	9	λn	λn	NOUN
ejde-942	387	10	0	0	NUM
ejde-942	387	11	4	4	NUM
ejde-942	387	12	4	4	NUM
ejde-942	387	13	16	16	NUM
ejde-942	387	14	16	16	NUM
ejde-942	387	15	36	36	NUM
ejde-942	387	16	36	36	NUM
ejde-942	387	17	64	64	NUM
ejde-942	387	18	64	64	NUM
ejde-942	387	19	100	100	NUM
ejde-942	387	20	λ̂n	λ̂n	PRON
ejde-942	387	21	0.0000	0.0000	NUM
ejde-942	387	22	4.0000	4.0000	NUM
ejde-942	387	23	3.9999	3.9999	NUM
ejde-942	387	24	15.9998	15.9998	NUM
ejde-942	387	25	16.0000	16.0000	NUM
ejde-942	387	26	36.0002	36.0002	NUM
ejde-942	387	27	36.0001	36.0001	NUM
ejde-942	387	28	64.0007	64.0007	NUM
ejde-942	387	29	63.9998	63.9998	NUM
ejde-942	387	30	99.9998	99.9998	NUM
ejde-942	387	31	example	example	NOUN
ejde-942	387	32	5.3	5.3	NUM
ejde-942	387	33	.	.	PUNCT
ejde-942	388	1	2d	2d	PROPN
ejde-942	388	2	sturm	sturm	PROPN
ejde-942	388	3	-	-	PUNCT
ejde-942	388	4	liouville	liouville	NOUN
ejde-942	388	5	problem	problem	NOUN
ejde-942	388	6	with	with	ADP
ejde-942	388	7	robin	robin	PROPN
ejde-942	388	8	boundary	boundary	PROPN
ejde-942	388	9	value	value	NOUN
ejde-942	388	10	conditions	condition	NOUN
ejde-942	388	11	[	[	X
ejde-942	388	12	10	10	NUM
ejde-942	388	13	]	]	X
ejde-942	388	14	:	:	PUNCT
ejde-942	388	15	−∇(ρ(x	−∇(ρ(x	PROPN
ejde-942	388	16	,	,	PUNCT
ejde-942	388	17	y)∇φ(x	y)∇φ(x	PROPN
ejde-942	388	18	,	,	PUNCT
ejde-942	388	19	y	y	NOUN
ejde-942	388	20	)	)	PUNCT
ejde-942	388	21	)	)	PUNCT
ejde-942	389	1	=	=	SYM
ejde-942	389	2	λρ(x	λρ(x	X
ejde-942	389	3	,	,	PUNCT
ejde-942	389	4	y)φ(x	y)φ(x	PROPN
ejde-942	389	5	,	,	PUNCT
ejde-942	389	6	y	y	PROPN
ejde-942	389	7	)	)	PUNCT
ejde-942	389	8	,	,	PUNCT
ejde-942	389	9	(	(	PUNCT
ejde-942	389	10	x	x	X
ejde-942	389	11	,	,	PUNCT
ejde-942	389	12	y	y	NOUN
ejde-942	389	13	)	)	PUNCT
ejde-942	389	14	∈	∈	PROPN
ejde-942	390	1	ω	ω	NUM
ejde-942	390	2	φ(0	φ(0	PROPN
ejde-942	390	3	,	,	PUNCT
ejde-942	390	4	y	y	NOUN
ejde-942	390	5	)	)	PUNCT
ejde-942	390	6	=	=	SYM
ejde-942	390	7	φx(π	φx(π	X
ejde-942	390	8	,	,	PUNCT
ejde-942	390	9	y	y	NOUN
ejde-942	390	10	)	)	PUNCT
ejde-942	390	11	=	=	SYM
ejde-942	390	12	φ(x	φ(x	NOUN
ejde-942	390	13	,	,	PUNCT
ejde-942	390	14	0	0	NUM
ejde-942	390	15	)	)	PUNCT
ejde-942	390	16	=	=	NOUN
ejde-942	390	17	φy(x	φy(x	X
ejde-942	390	18	,	,	PUNCT
ejde-942	390	19	π	π	X
ejde-942	390	20	)	)	PUNCT
ejde-942	390	21	=	=	SYM
ejde-942	391	1	0	0	NUM
ejde-942	391	2	,	,	PUNCT
ejde-942	391	3	(	(	PUNCT
ejde-942	391	4	5.3	5.3	NUM
ejde-942	391	5	)	)	PUNCT
ejde-942	391	6	where	where	SCONJ
ejde-942	391	7	ρ(x	ρ(x	NOUN
ejde-942	391	8	,	,	PUNCT
ejde-942	391	9	y	y	NOUN
ejde-942	391	10	)	)	PUNCT
ejde-942	391	11	=	=	PUNCT
ejde-942	392	1	(	(	PUNCT
ejde-942	392	2	1	1	NUM
ejde-942	392	3	+	+	NUM
ejde-942	392	4	0.3	0.3	NUM
ejde-942	392	5	sinx)(1	sinx)(1	NOUN
ejde-942	392	6	+	+	CCONJ
ejde-942	392	7	0.3	0.3	NUM
ejde-942	392	8	sin	sin	NOUN
ejde-942	392	9	y	y	PROPN
ejde-942	392	10	)	)	PUNCT
ejde-942	392	11	.	.	PUNCT
ejde-942	393	1	in	in	ADP
ejde-942	393	2	this	this	DET
ejde-942	393	3	example	example	NOUN
ejde-942	393	4	,	,	PUNCT
ejde-942	393	5	the	the	DET
ejde-942	393	6	finite	finite	PROPN
ejde-942	393	7	element	element	NOUN
ejde-942	393	8	method	method	NOUN
ejde-942	393	9	is	be	AUX
ejde-942	393	10	used	use	VERB
ejde-942	393	11	as	as	ADP
ejde-942	393	12	the	the	DET
ejde-942	393	13	baseline	baseline	NOUN
ejde-942	393	14	to	to	PART
ejde-942	393	15	compare	compare	VERB
ejde-942	393	16	our	our	PRON
ejde-942	393	17	neural	neural	ADJ
ejde-942	393	18	network	network	NOUN
ejde-942	393	19	method	method	NOUN
ejde-942	393	20	.	.	PUNCT
ejde-942	394	1	the	the	DET
ejde-942	394	2	number	number	NOUN
ejde-942	394	3	of	of	ADP
ejde-942	394	4	finite	finite	ADJ
ejde-942	394	5	elements	element	NOUN
ejde-942	394	6	is	be	AUX
ejde-942	394	7	set	set	VERB
ejde-942	394	8	as	as	ADP
ejde-942	394	9	47905	47905	NUM
ejde-942	394	10	by	by	ADP
ejde-942	394	11	using	use	VERB
ejde-942	394	12	the	the	DET
ejde-942	394	13	software	software	NOUN
ejde-942	394	14	of	of	ADP
ejde-942	394	15	freefem++	freefem++	PROPN
ejde-942	394	16	.	.	PUNCT
ejde-942	395	1	usually	usually	ADV
ejde-942	395	2	,	,	PUNCT
ejde-942	395	3	the	the	DET
ejde-942	395	4	finite	finite	ADJ
ejde-942	395	5	elements	element	NOUN
ejde-942	395	6	method	method	NOUN
ejde-942	395	7	has	have	VERB
ejde-942	395	8	very	very	ADV
ejde-942	395	9	high	high	ADJ
ejde-942	395	10	accuracy	accuracy	NOUN
ejde-942	395	11	.	.	PUNCT
ejde-942	396	1	in	in	ADP
ejde-942	396	2	table	table	NOUN
ejde-942	396	3	5.3	5.3	NUM
ejde-942	396	4	,	,	PUNCT
ejde-942	396	5	λ∗n	λ∗n	NUM
ejde-942	396	6	represents	represent	VERB
ejde-942	396	7	the	the	DET
ejde-942	396	8	numerical	numerical	ADJ
ejde-942	396	9	eigenvalues	eigenvalue	NOUN
ejde-942	396	10	obtained	obtain	VERB
ejde-942	396	11	by	by	ADP
ejde-942	396	12	finite	finite	ADJ
ejde-942	396	13	element	element	NOUN
ejde-942	396	14	method	method	NOUN
ejde-942	396	15	and	and	CCONJ
ejde-942	396	16	λ̂n	λ̂n	PUNCT
ejde-942	396	17	represents	represent	VERB
ejde-942	396	18	the	the	DET
ejde-942	396	19	numerical	numerical	ADJ
ejde-942	396	20	results	result	NOUN
ejde-942	396	21	obtained	obtain	VERB
ejde-942	396	22	from	from	ADP
ejde-942	396	23	our	our	PRON
ejde-942	396	24	deep	deep	ADJ
ejde-942	396	25	learning	learning	NOUN
ejde-942	396	26	algorithm	algorithm	NOUN
ejde-942	396	27	.	.	PUNCT
ejde-942	397	1	in	in	ADP
ejde-942	397	2	figure	figure	NOUN
ejde-942	397	3	5.2	5.2	NUM
ejde-942	397	4	,	,	PUNCT
ejde-942	397	5	we	we	PRON
ejde-942	397	6	show	show	VERB
ejde-942	397	7	the	the	DET
ejde-942	397	8	mutual	mutual	ADJ
ejde-942	397	9	orthogonal	orthogonal	ADJ
ejde-942	397	10	eigenfunctions	eigenfunction	NOUN
ejde-942	397	11	that	that	PRON
ejde-942	397	12	are	be	AUX
ejde-942	397	13	learned	learn	VERB
ejde-942	397	14	simultaneously	simultaneously	ADV
ejde-942	397	15	with	with	ADP
ejde-942	397	16	eigenvalues	eigenvalue	NOUN
ejde-942	397	17	.	.	PUNCT
ejde-942	398	1	such	such	ADJ
ejde-942	398	2	eigenfunctions	eigenfunction	NOUN
ejde-942	398	3	are	be	AUX
ejde-942	398	4	hard	hard	ADJ
ejde-942	398	5	to	to	PART
ejde-942	398	6	learn	learn	VERB
ejde-942	398	7	for	for	ADP
ejde-942	398	8	a	a	DET
ejde-942	398	9	non	non	ADJ
ejde-942	398	10	-	-	ADJ
ejde-942	398	11	expert	expert	ADJ
ejde-942	398	12	using	use	VERB
ejde-942	398	13	classical	classical	ADJ
ejde-942	398	14	numerical	numerical	ADJ
ejde-942	398	15	methods	method	NOUN
ejde-942	398	16	.	.	PUNCT
ejde-942	399	1	but	but	CCONJ
ejde-942	399	2	our	our	PRON
ejde-942	399	3	method	method	NOUN
ejde-942	399	4	is	be	AUX
ejde-942	399	5	beginner	beginner	NOUN
ejde-942	399	6	-	-	PUNCT
ejde-942	399	7	friendly	friendly	ADJ
ejde-942	399	8	.	.	PUNCT
ejde-942	399	9	table	table	NOUN
ejde-942	399	10	5.3	5.3	NUM
ejde-942	399	11	.	.	PUNCT
ejde-942	400	1	eigenvalues	eigenvalue	NOUN
ejde-942	400	2	of	of	ADP
ejde-942	400	3	2d	2d	NUM
ejde-942	400	4	sturm	sturm	PROPN
ejde-942	400	5	-	-	PUNCT
ejde-942	400	6	liouville	liouville	NOUN
ejde-942	400	7	problem	problem	NOUN
ejde-942	400	8	(	(	PUNCT
ejde-942	400	9	5.3	5.3	NUM
ejde-942	400	10	)	)	PUNCT
ejde-942	400	11	λn	λn	NOUN
ejde-942	400	12	λ1	λ1	PROPN
ejde-942	400	13	λ2	λ2	NOUN
ejde-942	401	1	λ3	λ3	PROPN
ejde-942	401	2	λ4	λ4	PROPN
ejde-942	401	3	λ5	λ5	VERB
ejde-942	401	4	λ6	λ6	PROPN
ejde-942	401	5	λ7	λ7	PROPN
ejde-942	401	6	λ8	λ8	PROPN
ejde-942	401	7	λ9	λ9	PROPN
ejde-942	401	8	λ10	λ10	NOUN
ejde-942	401	9	λ∗	λ∗	PROPN
ejde-942	401	10	n	n	CCONJ
ejde-942	402	1	0.4959	0.4959	NUM
ejde-942	402	2	2.5067	2.5067	NUM
ejde-942	402	3	2.5067	2.5067	NUM
ejde-942	402	4	4.5175	4.5175	NUM
ejde-942	402	5	6.5069	6.5069	NUM
ejde-942	402	6	6.5069	6.5069	NUM
ejde-942	402	7	8.5177	8.5177	NUM
ejde-942	402	8	8.5177	8.5177	NUM
ejde-942	402	9	12.5069	12.5069	NUM
ejde-942	402	10	12.5069	12.5069	NUM
ejde-942	402	11	λ̂n	λ̂n	PRON
ejde-942	402	12	0.4959	0.4959	NUM
ejde-942	402	13	2.5067	2.5067	NUM
ejde-942	402	14	2.5067	2.5067	NUM
ejde-942	402	15	4.5174	4.5174	NUM
ejde-942	402	16	6.5068	6.5068	NUM
ejde-942	402	17	6.5068	6.5068	NUM
ejde-942	402	18	8.5176	8.5176	NUM
ejde-942	402	19	8.5176	8.5176	NUM
ejde-942	402	20	12.5075	12.5075	NUM
ejde-942	402	21	12.5071	12.5071	NUM
ejde-942	402	22	the	the	DET
ejde-942	402	23	final	final	ADJ
ejde-942	402	24	example	example	NOUN
ejde-942	402	25	demonstrates	demonstrate	VERB
ejde-942	402	26	that	that	SCONJ
ejde-942	402	27	our	our	PRON
ejde-942	402	28	neural	neural	ADJ
ejde-942	402	29	network	network	NOUN
ejde-942	402	30	possesses	possess	VERB
ejde-942	402	31	considerably	considerably	ADV
ejde-942	402	32	good	good	ADJ
ejde-942	402	33	accuracy	accuracy	NOUN
ejde-942	402	34	.	.	PUNCT
ejde-942	403	1	our	our	PRON
ejde-942	403	2	neural	neural	ADJ
ejde-942	403	3	network	network	NOUN
ejde-942	403	4	can	can	AUX
ejde-942	403	5	be	be	AUX
ejde-942	403	6	easily	easily	ADV
ejde-942	403	7	utilized	utilize	VERB
ejde-942	403	8	,	,	PUNCT
ejde-942	403	9	whereas	whereas	SCONJ
ejde-942	403	10	the	the	DET
ejde-942	403	11	finite	finite	PROPN
ejde-942	403	12	element	element	NOUN
ejde-942	403	13	method	method	NOUN
ejde-942	403	14	poses	pose	VERB
ejde-942	403	15	significant	significant	ADJ
ejde-942	403	16	difficulties	difficulty	NOUN
ejde-942	403	17	for	for	ADP
ejde-942	403	18	non	non	NOUN
ejde-942	403	19	-	-	NOUN
ejde-942	403	20	experts	expert	NOUN
ejde-942	403	21	.	.	PUNCT
ejde-942	404	1	how	how	SCONJ
ejde-942	404	2	to	to	PART
ejde-942	404	3	further	far	ADV
ejde-942	404	4	improve	improve	VERB
ejde-942	404	5	accuracy	accuracy	NOUN
ejde-942	404	6	of	of	ADP
ejde-942	404	7	neural	neural	ADJ
ejde-942	404	8	network	network	NOUN
ejde-942	404	9	methods	method	NOUN
ejde-942	404	10	remains	remain	VERB
ejde-942	404	11	an	an	DET
ejde-942	404	12	important	important	ADJ
ejde-942	404	13	question	question	NOUN
ejde-942	404	14	for	for	ADP
ejde-942	404	15	future	future	ADJ
ejde-942	404	16	studies	study	NOUN
ejde-942	404	17	.	.	PUNCT
ejde-942	405	1	6	6	X
ejde-942	405	2	.	.	X
ejde-942	405	3	conclusion	conclusion	NOUN
ejde-942	405	4	deep	deep	ADJ
ejde-942	405	5	learning	learning	NOUN
ejde-942	405	6	techniques	technique	NOUN
ejde-942	405	7	have	have	AUX
ejde-942	405	8	shown	show	VERB
ejde-942	405	9	significant	significant	ADJ
ejde-942	405	10	advancements	advancement	NOUN
ejde-942	405	11	in	in	ADP
ejde-942	405	12	addressing	address	VERB
ejde-942	405	13	eigenvalue	eigenvalue	NOUN
ejde-942	405	14	problems	problem	NOUN
ejde-942	405	15	,	,	PUNCT
ejde-942	405	16	particularly	particularly	ADV
ejde-942	405	17	with	with	ADP
ejde-942	405	18	the	the	DET
ejde-942	405	19	use	use	NOUN
ejde-942	405	20	of	of	ADP
ejde-942	405	21	pinn	pinn	NOUN
ejde-942	405	22	(	(	PUNCT
ejde-942	405	23	physics	physics	NOUN
ejde-942	405	24	-	-	PUNCT
ejde-942	405	25	informed	inform	VERB
ejde-942	405	26	neural	neural	ADJ
ejde-942	405	27	networks	network	NOUN
ejde-942	405	28	)	)	PUNCT
ejde-942	405	29	due	due	ADP
ejde-942	405	30	to	to	ADP
ejde-942	405	31	its	its	PRON
ejde-942	405	32	ease	ease	NOUN
ejde-942	405	33	of	of	ADP
ejde-942	405	34	implementation	implementation	NOUN
ejde-942	405	35	and	and	CCONJ
ejde-942	405	36	ability	ability	NOUN
ejde-942	405	37	to	to	PART
ejde-942	405	38	maintain	maintain	VERB
ejde-942	405	39	operator	operator	NOUN
ejde-942	405	40	selfadjointness	selfadjointness	NOUN
ejde-942	405	41	.	.	PUNCT
ejde-942	406	1	pinn	pinn	PROPN
ejde-942	406	2	has	have	AUX
ejde-942	406	3	been	be	AUX
ejde-942	406	4	applied	apply	VERB
ejde-942	406	5	to	to	PART
ejde-942	406	6	solve	solve	VERB
ejde-942	406	7	quantum	quantum	NOUN
ejde-942	406	8	problems	problem	NOUN
ejde-942	406	9	and	and	CCONJ
ejde-942	406	10	sturm	sturm	PROPN
ejde-942	406	11	-	-	PUNCT
ejde-942	406	12	liouville	liouville	NOUN
ejde-942	406	13	16	16	NUM
ejde-942	406	14	s.	s.	PROPN
ejde-942	406	15	zhang	zhang	PROPN
ejde-942	406	16	,	,	PUNCT
ejde-942	406	17	j.	j.	PROPN
ejde-942	406	18	zu	zu	PROPN
ejde-942	406	19	,	,	PUNCT
ejde-942	406	20	j.	j.	PROPN
ejde-942	406	21	zhang	zhang	PROPN
ejde-942	407	1	ejde-2024/53	ejde-2024/53	PROPN
ejde-942	407	2	figure	figure	NOUN
ejde-942	407	3	5.2	5.2	NUM
ejde-942	407	4	.	.	PUNCT
ejde-942	408	1	eigenfunctions	eigenfunction	NOUN
ejde-942	408	2	of	of	ADP
ejde-942	408	3	2d	2d	PROPN
ejde-942	408	4	sturm	sturm	NOUN
ejde-942	408	5	-	-	PUNCT
ejde-942	408	6	liouville	liouville	NOUN
ejde-942	408	7	problem	problem	NOUN
ejde-942	408	8	with	with	ADP
ejde-942	408	9	(	(	PUNCT
ejde-942	408	10	5.3	5.3	NUM
ejde-942	408	11	)	)	PUNCT
ejde-942	408	12	boundary	boundary	ADJ
ejde-942	408	13	value	value	NOUN
ejde-942	408	14	problems	problem	NOUN
ejde-942	408	15	,	,	PUNCT
ejde-942	408	16	but	but	CCONJ
ejde-942	408	17	with	with	ADP
ejde-942	408	18	limitations	limitation	NOUN
ejde-942	408	19	such	such	ADJ
ejde-942	408	20	as	as	ADP
ejde-942	408	21	one	one	NUM
ejde-942	408	22	-	-	PUNCT
ejde-942	408	23	dimensional	dimensional	ADJ
ejde-942	408	24	homogeneous	homogeneous	ADJ
ejde-942	408	25	media	medium	NOUN
ejde-942	408	26	or	or	CCONJ
ejde-942	408	27	specific	specific	ADJ
ejde-942	408	28	boundary	boundary	ADJ
ejde-942	408	29	value	value	NOUN
ejde-942	408	30	conditions	condition	NOUN
ejde-942	408	31	.	.	PUNCT
ejde-942	409	1	this	this	DET
ejde-942	409	2	paper	paper	NOUN
ejde-942	409	3	introduces	introduce	VERB
ejde-942	409	4	a	a	DET
ejde-942	409	5	novel	novel	ADJ
ejde-942	409	6	deep	deep	ADJ
ejde-942	409	7	learning	learning	NOUN
ejde-942	409	8	method	method	NOUN
ejde-942	409	9	to	to	PART
ejde-942	409	10	find	find	VERB
ejde-942	409	11	the	the	DET
ejde-942	409	12	smallest	small	ADJ
ejde-942	409	13	eigenpairs	eigenpair	NOUN
ejde-942	409	14	by	by	ADP
ejde-942	409	15	constructing	construct	VERB
ejde-942	409	16	a	a	DET
ejde-942	409	17	new	new	ADJ
ejde-942	409	18	cost	cost	NOUN
ejde-942	409	19	function	function	NOUN
ejde-942	409	20	,	,	PUNCT
ejde-942	409	21	incorporating	incorporate	VERB
ejde-942	409	22	adaptive	adaptive	ADJ
ejde-942	409	23	hyper	hyper	ADJ
ejde-942	409	24	-	-	ADJ
ejde-942	409	25	parameter	parameter	ADJ
ejde-942	409	26	tuning	tuning	NOUN
ejde-942	409	27	,	,	PUNCT
ejde-942	409	28	and	and	CCONJ
ejde-942	409	29	sequentially	sequentially	ADV
ejde-942	409	30	training	training	NOUN
ejde-942	409	31	eigenpairs	eigenpair	NOUN
ejde-942	409	32	.	.	PUNCT
ejde-942	410	1	the	the	DET
ejde-942	410	2	proposed	propose	VERB
ejde-942	410	3	approach	approach	NOUN
ejde-942	410	4	extends	extend	VERB
ejde-942	410	5	to	to	ADP
ejde-942	410	6	varying	vary	VERB
ejde-942	410	7	boundary	boundary	ADJ
ejde-942	410	8	value	value	NOUN
ejde-942	410	9	conditions	condition	NOUN
ejde-942	410	10	in	in	ADP
ejde-942	410	11	inhomogeneous	inhomogeneous	ADJ
ejde-942	410	12	media	medium	NOUN
ejde-942	410	13	,	,	PUNCT
ejde-942	410	14	two	two	NUM
ejde-942	410	15	-	-	PUNCT
ejde-942	410	16	dimensional	dimensional	ADJ
ejde-942	410	17	cases	case	NOUN
ejde-942	410	18	,	,	PUNCT
ejde-942	410	19	and	and	CCONJ
ejde-942	410	20	periodic	periodic	ADJ
ejde-942	410	21	scenarios	scenario	NOUN
ejde-942	410	22	,	,	PUNCT
ejde-942	410	23	demonstrating	demonstrate	VERB
ejde-942	410	24	improved	improved	ADJ
ejde-942	410	25	simplicity	simplicity	NOUN
ejde-942	410	26	,	,	PUNCT
ejde-942	410	27	accuracy	accuracy	NOUN
ejde-942	410	28	,	,	PUNCT
ejde-942	410	29	and	and	CCONJ
ejde-942	410	30	interpretability	interpretability	NOUN
ejde-942	410	31	.	.	PUNCT
ejde-942	411	1	quantitative	quantitative	ADJ
ejde-942	411	2	estimation	estimation	NOUN
ejde-942	411	3	of	of	ADP
ejde-942	411	4	eigenpairs	eigenpair	NOUN
ejde-942	411	5	is	be	AUX
ejde-942	411	6	given	give	VERB
ejde-942	411	7	for	for	ADP
ejde-942	411	8	the	the	DET
ejde-942	411	9	sturm	sturm	NOUN
ejde-942	411	10	-	-	PUNCT
ejde-942	411	11	liouville	liouville	NOUN
ejde-942	411	12	eigenvalue	eigenvalue	NOUN
ejde-942	411	13	problems	problem	NOUN
ejde-942	411	14	.	.	PUNCT
ejde-942	412	1	numerical	numerical	ADJ
ejde-942	412	2	experiments	experiment	NOUN
ejde-942	412	3	validate	validate	VERB
ejde-942	412	4	the	the	DET
ejde-942	412	5	method	method	NOUN
ejde-942	412	6	’s	’s	PART
ejde-942	412	7	efficiency	efficiency	NOUN
ejde-942	412	8	and	and	CCONJ
ejde-942	412	9	accuracy	accuracy	NOUN
ejde-942	412	10	compared	compare	VERB
ejde-942	412	11	to	to	ADP
ejde-942	412	12	previous	previous	ADJ
ejde-942	412	13	algorithms	algorithm	NOUN
ejde-942	412	14	[	[	X
ejde-942	412	15	5	5	NUM
ejde-942	412	16	]	]	PUNCT
ejde-942	412	17	.	.	PUNCT
ejde-942	413	1	references	reference	NOUN
ejde-942	413	2	[	[	X
ejde-942	413	3	1	1	NUM
ejde-942	413	4	]	]	PUNCT
ejde-942	413	5	r.	r.	PROPN
ejde-942	413	6	anderssen	anderssen	PROPN
ejde-942	413	7	,	,	PUNCT
ejde-942	413	8	f.	f.	PROPN
ejde-942	413	9	de	de	PROPN
ejde-942	413	10	hoog	hoog	PROPN
ejde-942	413	11	;	;	PUNCT
ejde-942	413	12	on	on	ADP
ejde-942	413	13	the	the	DET
ejde-942	413	14	correction	correction	NOUN
ejde-942	413	15	of	of	ADP
ejde-942	413	16	finite	finite	ADJ
ejde-942	413	17	difference	difference	NOUN
ejde-942	413	18	eigenvalue	eigenvalue	NOUN
ejde-942	413	19	approximations	approximation	NOUN
ejde-942	413	20	for	for	ADP
ejde-942	413	21	sturm	sturm	NOUN
ejde-942	413	22	-	-	PUNCT
ejde-942	413	23	liouville	liouville	NOUN
ejde-942	413	24	problems	problem	NOUN
ejde-942	413	25	with	with	ADP
ejde-942	413	26	general	general	ADJ
ejde-942	413	27	boundary	boundary	ADJ
ejde-942	413	28	conditions	condition	NOUN
ejde-942	413	29	,	,	PUNCT
ejde-942	413	30	bit	bit	NOUN
ejde-942	413	31	numerical	numerical	ADJ
ejde-942	413	32	mathematics	mathematic	NOUN
ejde-942	413	33	,	,	PUNCT
ejde-942	413	34	24	24	NUM
ejde-942	413	35	(	(	PUNCT
ejde-942	413	36	1984	1984	NUM
ejde-942	413	37	)	)	PUNCT
ejde-942	413	38	,	,	PUNCT
ejde-942	413	39	401–412	401–412	NUM
ejde-942	413	40	.	.	PUNCT
ejde-942	414	1	[	[	X
ejde-942	414	2	2	2	NUM
ejde-942	414	3	]	]	PUNCT
ejde-942	414	4	a.	a.	PROPN
ejde-942	414	5	andrew	andrew	PROPN
ejde-942	414	6	,	,	PUNCT
ejde-942	414	7	j.	j.	PROPN
ejde-942	414	8	paine	paine	PROPN
ejde-942	414	9	;	;	PUNCT
ejde-942	414	10	correction	correction	NOUN
ejde-942	414	11	of	of	ADP
ejde-942	414	12	finite	finite	PROPN
ejde-942	414	13	element	element	NOUN
ejde-942	414	14	estimates	estimate	NOUN
ejde-942	414	15	for	for	ADP
ejde-942	414	16	sturm	sturm	NOUN
ejde-942	414	17	-	-	PUNCT
ejde-942	414	18	liouville	liouville	NOUN
ejde-942	414	19	eigenvalues	eigenvalue	NOUN
ejde-942	414	20	,	,	PUNCT
ejde-942	414	21	numerische	numerische	PROPN
ejde-942	414	22	mathematik	mathematik	PROPN
ejde-942	414	23	,	,	PUNCT
ejde-942	414	24	50	50	NUM
ejde-942	414	25	(	(	PUNCT
ejde-942	414	26	1986	1986	NUM
ejde-942	414	27	)	)	PUNCT
ejde-942	414	28	,	,	PUNCT
ejde-942	414	29	205–215	205–215	NUM
ejde-942	414	30	.	.	PUNCT
ejde-942	415	1	[	[	X
ejde-942	415	2	3	3	X
ejde-942	415	3	]	]	X
ejde-942	415	4	v.	v.	CCONJ
ejde-942	415	5	barbu	barbu	PROPN
ejde-942	415	6	,	,	PUNCT
ejde-942	415	7	n.	n.	PROPN
ejde-942	415	8	pavel	pavel	PROPN
ejde-942	415	9	;	;	PUNCT
ejde-942	415	10	periodic	periodic	ADJ
ejde-942	415	11	solutions	solution	NOUN
ejde-942	415	12	to	to	PART
ejde-942	415	13	nonlinear	nonlinear	VERB
ejde-942	415	14	one	one	NUM
ejde-942	415	15	dimensional	dimensional	ADJ
ejde-942	415	16	wave	wave	NOUN
ejde-942	415	17	equation	equation	NOUN
ejde-942	415	18	with	with	ADP
ejde-942	415	19	x	x	ADJ
ejde-942	415	20	-	-	ADJ
ejde-942	415	21	dependent	dependent	ADJ
ejde-942	415	22	coefficients	coefficient	NOUN
ejde-942	415	23	,	,	PUNCT
ejde-942	415	24	transactions	transaction	NOUN
ejde-942	415	25	of	of	ADP
ejde-942	415	26	the	the	DET
ejde-942	415	27	american	american	PROPN
ejde-942	415	28	mathematical	mathematical	PROPN
ejde-942	415	29	society	society	NOUN
ejde-942	415	30	,	,	PUNCT
ejde-942	415	31	349	349	NUM
ejde-942	415	32	(	(	PUNCT
ejde-942	415	33	1997	1997	NUM
ejde-942	415	34	)	)	PUNCT
ejde-942	415	35	,	,	PUNCT
ejde-942	415	36	2035–2048	2035–2048	NUM
ejde-942	415	37	.	.	PUNCT
ejde-942	416	1	[	[	X
ejde-942	416	2	4	4	X
ejde-942	416	3	]	]	PUNCT
ejde-942	416	4	k.	k.	PROPN
ejde-942	416	5	beauchard	beauchard	PROPN
ejde-942	416	6	,	,	PUNCT
ejde-942	416	7	m.	m.	NOUN
ejde-942	416	8	mirrahimi	mirrahimi	PROPN
ejde-942	416	9	;	;	PUNCT
ejde-942	416	10	practical	practical	ADJ
ejde-942	416	11	stabilization	stabilization	NOUN
ejde-942	416	12	of	of	ADP
ejde-942	416	13	a	a	DET
ejde-942	416	14	quantum	quantum	NOUN
ejde-942	416	15	particle	particle	NOUN
ejde-942	416	16	in	in	ADP
ejde-942	416	17	a	a	DET
ejde-942	416	18	onedimensional	onedimensional	ADJ
ejde-942	416	19	infinite	infinite	ADJ
ejde-942	416	20	square	square	ADJ
ejde-942	416	21	potential	potential	NOUN
ejde-942	416	22	well	well	ADV
ejde-942	416	23	,	,	PUNCT
ejde-942	416	24	siam	siam	ADJ
ejde-942	416	25	journal	journal	NOUN
ejde-942	416	26	on	on	ADP
ejde-942	416	27	control	control	NOUN
ejde-942	416	28	and	and	CCONJ
ejde-942	416	29	optimization	optimization	NOUN
ejde-942	416	30	,	,	PUNCT
ejde-942	416	31	48	48	NUM
ejde-942	416	32	(	(	PUNCT
ejde-942	416	33	2009	2009	NUM
ejde-942	416	34	)	)	PUNCT
ejde-942	416	35	,	,	PUNCT
ejde-942	416	36	1179–1205	1179–1205	NUM
ejde-942	416	37	.	.	PUNCT
ejde-942	417	1	[	[	X
ejde-942	417	2	5	5	NUM
ejde-942	417	3	]	]	X
ejde-942	417	4	i.	i.	PROPN
ejde-942	417	5	ben	ben	PROPN
ejde-942	417	6	-	-	PUNCT
ejde-942	417	7	shaul	shaul	PROPN
ejde-942	417	8	,	,	PUNCT
ejde-942	417	9	l.	l.	PROPN
ejde-942	417	10	bar	bar	PROPN
ejde-942	417	11	,	,	PUNCT
ejde-942	417	12	d.	d.	PROPN
ejde-942	417	13	fishelov	fishelov	PROPN
ejde-942	417	14	,	,	PUNCT
ejde-942	417	15	n.	n.	NOUN
ejde-942	417	16	sochen	sochen	NOUN
ejde-942	417	17	;	;	PUNCT
ejde-942	417	18	deep	deep	ADJ
ejde-942	417	19	learning	learning	NOUN
ejde-942	417	20	solution	solution	NOUN
ejde-942	417	21	of	of	ADP
ejde-942	417	22	the	the	DET
ejde-942	417	23	eigenvalue	eigenvalue	PROPN
ejde-942	417	24	problem	problem	NOUN
ejde-942	417	25	for	for	ADP
ejde-942	417	26	differential	differential	ADJ
ejde-942	417	27	operators	operator	NOUN
ejde-942	417	28	,	,	PUNCT
ejde-942	417	29	neural	neural	ADJ
ejde-942	417	30	computation	computation	NOUN
ejde-942	417	31	,	,	PUNCT
ejde-942	417	32	35	35	NUM
ejde-942	417	33	(	(	PUNCT
ejde-942	417	34	2023	2023	NUM
ejde-942	417	35	)	)	PUNCT
ejde-942	417	36	,	,	PUNCT
ejde-942	417	37	1100–1134	1100–1134	NUM
ejde-942	417	38	.	.	PUNCT
ejde-942	418	1	[	[	X
ejde-942	418	2	6	6	NUM
ejde-942	418	3	]	]	PUNCT
ejde-942	418	4	r.	r.	PROPN
ejde-942	418	5	chen	chen	PROPN
ejde-942	418	6	,	,	PUNCT
ejde-942	418	7	y.	y.	PROPN
ejde-942	418	8	rubanova	rubanova	PROPN
ejde-942	418	9	,	,	PUNCT
ejde-942	418	10	j.	j.	PROPN
ejde-942	418	11	bettencourt	bettencourt	PROPN
ejde-942	418	12	,	,	PUNCT
ejde-942	418	13	d.	d.	PROPN
ejde-942	418	14	duvenaud	duvenaud	PROPN
ejde-942	418	15	;	;	PUNCT
ejde-942	418	16	neural	neural	ADJ
ejde-942	418	17	ordinary	ordinary	ADJ
ejde-942	418	18	differential	differential	ADJ
ejde-942	418	19	equations	equation	NOUN
ejde-942	418	20	,	,	PUNCT
ejde-942	418	21	neurips	neurip	NOUN
ejde-942	418	22	2018	2018	NUM
ejde-942	418	23	,	,	PUNCT
ejde-942	418	24	31	31	NUM
ejde-942	418	25	(	(	PUNCT
ejde-942	418	26	2018	2018	NUM
ejde-942	418	27	)	)	PUNCT
ejde-942	418	28	.	.	PUNCT
ejde-942	419	1	[	[	X
ejde-942	419	2	7	7	X
ejde-942	419	3	]	]	X
ejde-942	419	4	s.	s.	PROPN
ejde-942	419	5	greydanus	greydanus	PROPN
ejde-942	419	6	,	,	PUNCT
ejde-942	419	7	m.	m.	NOUN
ejde-942	419	8	dzamba	dzamba	NOUN
ejde-942	419	9	,	,	PUNCT
ejde-942	419	10	j.	j.	PROPN
ejde-942	419	11	yosinski	yosinski	PROPN
ejde-942	419	12	;	;	PUNCT
ejde-942	419	13	hamiltonian	hamiltonian	ADJ
ejde-942	419	14	neural	neural	ADJ
ejde-942	419	15	networks	network	NOUN
ejde-942	419	16	,	,	PUNCT
ejde-942	419	17	neurips	neurip	NOUN
ejde-942	419	18	2019	2019	NUM
ejde-942	419	19	,	,	PUNCT
ejde-942	419	20	32	32	NUM
ejde-942	419	21	(	(	PUNCT
ejde-942	419	22	2019	2019	NUM
ejde-942	419	23	)	)	PUNCT
ejde-942	419	24	.	.	PUNCT
ejde-942	420	1	[	[	X
ejde-942	420	2	8	8	X
ejde-942	420	3	]	]	X
ejde-942	420	4	e.	e.	PROPN
ejde-942	420	5	holliday	holliday	PROPN
ejde-942	420	6	,	,	PUNCT
ejde-942	420	7	j.	j.	PROPN
ejde-942	420	8	lindner	lindner	PROPN
ejde-942	420	9	,	,	PUNCT
ejde-942	420	10	w.	w.	PROPN
ejde-942	420	11	ditto	ditto	NOUN
ejde-942	420	12	;	;	PUNCT
ejde-942	420	13	solving	solve	VERB
ejde-942	420	14	quantum	quantum	ADJ
ejde-942	420	15	billiard	billiard	NOUN
ejde-942	420	16	eigenvalue	eigenvalue	NOUN
ejde-942	420	17	problems	problem	NOUN
ejde-942	420	18	with	with	ADP
ejde-942	420	19	physicsinformed	physicsinforme	VERB
ejde-942	420	20	machine	machine	NOUN
ejde-942	420	21	learning	learning	NOUN
ejde-942	420	22	,	,	PUNCT
ejde-942	420	23	aip	aip	PROPN
ejde-942	420	24	advances	advance	NOUN
ejde-942	420	25	,	,	PUNCT
ejde-942	420	26	13	13	NUM
ejde-942	420	27	(	(	PUNCT
ejde-942	420	28	2023	2023	NUM
ejde-942	420	29	)	)	PUNCT
ejde-942	420	30	,	,	PUNCT
ejde-942	420	31	085013	085013	NUM
ejde-942	420	32	.	.	PUNCT
ejde-942	421	1	[	[	X
ejde-942	421	2	9	9	NUM
ejde-942	421	3	]	]	PUNCT
ejde-942	421	4	h.	h.	PROPN
ejde-942	421	5	jin	jin	PROPN
ejde-942	421	6	,	,	PUNCT
ejde-942	421	7	m.	m.	NOUN
ejde-942	421	8	mattheakis	mattheakis	PROPN
ejde-942	421	9	,	,	PUNCT
ejde-942	421	10	p.	p.	PROPN
ejde-942	421	11	protopapas	protopapas	PROPN
ejde-942	421	12	;	;	PUNCT
ejde-942	421	13	physics	physics	NOUN
ejde-942	421	14	-	-	PUNCT
ejde-942	421	15	informed	inform	VERB
ejde-942	421	16	neural	neural	ADJ
ejde-942	421	17	networks	network	NOUN
ejde-942	421	18	for	for	ADP
ejde-942	421	19	quantum	quantum	NOUN
ejde-942	421	20	eigenvalue	eigenvalue	NOUN
ejde-942	421	21	problems	problem	NOUN
ejde-942	421	22	,	,	PUNCT
ejde-942	421	23	in	in	ADP
ejde-942	421	24	ijcnn	ijcnn	PROPN
ejde-942	421	25	at	at	ADP
ejde-942	421	26	ieee	ieee	PROPN
ejde-942	421	27	world	world	PROPN
ejde-942	421	28	congress	congress	PROPN
ejde-942	421	29	on	on	ADP
ejde-942	421	30	computational	computational	ADJ
ejde-942	421	31	intelligence	intelligence	NOUN
ejde-942	421	32	(	(	PUNCT
ejde-942	421	33	2022	2022	NUM
ejde-942	421	34	)	)	PUNCT
ejde-942	421	35	.	.	PUNCT
ejde-942	422	1	[	[	X
ejde-942	422	2	10	10	NUM
ejde-942	422	3	]	]	PUNCT
ejde-942	423	1	s.	s.	PROPN
ejde-942	423	2	ji	ji	PROPN
ejde-942	423	3	;	;	PUNCT
ejde-942	423	4	periodic	periodic	ADJ
ejde-942	423	5	solutions	solution	NOUN
ejde-942	423	6	of	of	ADP
ejde-942	423	7	two	two	NUM
ejde-942	423	8	-	-	PUNCT
ejde-942	423	9	dimensional	dimensional	ADJ
ejde-942	423	10	wave	wave	NOUN
ejde-942	423	11	equations	equation	NOUN
ejde-942	423	12	with	with	ADP
ejde-942	423	13	x	x	ADJ
ejde-942	423	14	-	-	ADJ
ejde-942	423	15	dependent	dependent	ADJ
ejde-942	423	16	coefficients	coefficient	NOUN
ejde-942	423	17	and	and	CCONJ
ejde-942	423	18	sturm	sturm	NOUN
ejde-942	423	19	-	-	PUNCT
ejde-942	423	20	liouville	liouville	VERB
ejde-942	423	21	boundary	boundary	ADJ
ejde-942	423	22	conditions	condition	NOUN
ejde-942	423	23	,	,	PUNCT
ejde-942	423	24	nonlinearity	nonlinearity	NOUN
ejde-942	423	25	,	,	PUNCT
ejde-942	423	26	35	35	NUM
ejde-942	423	27	(	(	PUNCT
ejde-942	423	28	2022	2022	NUM
ejde-942	423	29	)	)	PUNCT
ejde-942	423	30	,	,	PUNCT
ejde-942	423	31	5033	5033	NUM
ejde-942	423	32	.	.	PUNCT
ejde-942	424	1	[	[	X
ejde-942	424	2	11	11	NUM
ejde-942	424	3	]	]	PUNCT
ejde-942	424	4	s.	s.	PROPN
ejde-942	424	5	ji	ji	PROPN
ejde-942	424	6	,	,	PUNCT
ejde-942	424	7	y.	y.	PROPN
ejde-942	424	8	li	li	PROPN
ejde-942	424	9	;	;	PUNCT
ejde-942	424	10	periodic	periodic	ADJ
ejde-942	424	11	solutions	solution	NOUN
ejde-942	424	12	to	to	ADP
ejde-942	424	13	one	one	NUM
ejde-942	424	14	-	-	PUNCT
ejde-942	424	15	dimensional	dimensional	ADJ
ejde-942	424	16	wave	wave	NOUN
ejde-942	424	17	equation	equation	NOUN
ejde-942	424	18	with	with	ADP
ejde-942	424	19	x	x	ADJ
ejde-942	424	20	-	-	ADJ
ejde-942	424	21	dependent	dependent	ADJ
ejde-942	424	22	coefficients	coefficient	NOUN
ejde-942	424	23	,	,	PUNCT
ejde-942	424	24	journal	journal	NOUN
ejde-942	424	25	of	of	ADP
ejde-942	424	26	differential	differential	ADJ
ejde-942	424	27	equations	equation	NOUN
ejde-942	424	28	,	,	PUNCT
ejde-942	424	29	229	229	NUM
ejde-942	424	30	(	(	PUNCT
ejde-942	424	31	2006	2006	NUM
ejde-942	424	32	)	)	PUNCT
ejde-942	424	33	,	,	PUNCT
ejde-942	424	34	466–493	466–493	NUM
ejde-942	424	35	.	.	PUNCT
ejde-942	425	1	ejde-2024/53	ejde-2024/53	VERB
ejde-942	425	2	deep	deep	ADJ
ejde-942	425	3	learning	learning	NOUN
ejde-942	425	4	method	method	NOUN
ejde-942	425	5	for	for	ADP
ejde-942	425	6	finding	find	VERB
ejde-942	425	7	eigenpairs	eigenpair	NOUN
ejde-942	425	8	17	17	NUM
ejde-942	426	1	[	[	X
ejde-942	426	2	12	12	NUM
ejde-942	426	3	]	]	PUNCT
ejde-942	426	4	p.	p.	NOUN
ejde-942	426	5	jin	jin	PROPN
ejde-942	426	6	,	,	PUNCT
ejde-942	426	7	z.	z.	PROPN
ejde-942	426	8	zhang	zhang	PROPN
ejde-942	426	9	,	,	PUNCT
ejde-942	426	10	i.	i.	PROPN
ejde-942	426	11	kevrekidis	kevrekidis	PROPN
ejde-942	426	12	,	,	PUNCT
ejde-942	426	13	g.	g.	PROPN
ejde-942	426	14	karniadakis	karniadakis	PROPN
ejde-942	426	15	;	;	PUNCT
ejde-942	426	16	learning	learn	VERB
ejde-942	426	17	poisson	poisson	NOUN
ejde-942	426	18	systems	system	NOUN
ejde-942	426	19	and	and	CCONJ
ejde-942	426	20	trajectories	trajectory	NOUN
ejde-942	426	21	of	of	ADP
ejde-942	426	22	autonomous	autonomous	ADJ
ejde-942	426	23	systems	system	NOUN
ejde-942	426	24	via	via	ADP
ejde-942	426	25	poisson	poisson	PROPN
ejde-942	426	26	neural	neural	PROPN
ejde-942	426	27	networks	network	NOUN
ejde-942	426	28	,	,	PUNCT
ejde-942	426	29	ieee	ieee	NOUN
ejde-942	426	30	transactions	transaction	NOUN
ejde-942	426	31	on	on	ADP
ejde-942	426	32	neural	neural	ADJ
ejde-942	426	33	networks	network	NOUN
ejde-942	426	34	and	and	CCONJ
ejde-942	426	35	learning	learning	NOUN
ejde-942	426	36	systems	system	NOUN
ejde-942	426	37	,	,	PUNCT
ejde-942	426	38	34	34	NUM
ejde-942	426	39	(	(	PUNCT
ejde-942	426	40	2022	2022	NUM
ejde-942	426	41	)	)	PUNCT
ejde-942	426	42	,	,	PUNCT
ejde-942	426	43	8271–8283	8271–8283	NUM
ejde-942	426	44	.	.	PUNCT
ejde-942	427	1	[	[	X
ejde-942	427	2	13	13	NUM
ejde-942	427	3	]	]	PUNCT
ejde-942	427	4	p.	p.	NOUN
ejde-942	427	5	jin	jin	PROPN
ejde-942	427	6	,	,	PUNCT
ejde-942	427	7	z.	z.	PROPN
ejde-942	427	8	zhang	zhang	PROPN
ejde-942	427	9	,	,	PUNCT
ejde-942	427	10	a.	a.	PROPN
ejde-942	427	11	zhu	zhu	PROPN
ejde-942	427	12	,	,	PUNCT
ejde-942	427	13	y.	y.	PROPN
ejde-942	427	14	tang	tang	PROPN
ejde-942	427	15	,	,	PUNCT
ejde-942	427	16	g.	g.	PROPN
ejde-942	427	17	karniadakis	karniadakis	PROPN
ejde-942	427	18	;	;	PUNCT
ejde-942	427	19	sympnets	sympnet	NOUN
ejde-942	427	20	:	:	PUNCT
ejde-942	427	21	intrinsic	intrinsic	ADJ
ejde-942	427	22	structure	structure	NOUN
ejde-942	427	23	-	-	PUNCT
ejde-942	427	24	preserving	preserve	VERB
ejde-942	427	25	symplectic	symplectic	ADJ
ejde-942	427	26	networks	network	NOUN
ejde-942	427	27	for	for	ADP
ejde-942	427	28	identifying	identify	VERB
ejde-942	427	29	hamiltonian	hamiltonian	ADJ
ejde-942	427	30	systems	system	NOUN
ejde-942	427	31	,	,	PUNCT
ejde-942	427	32	neural	neural	ADJ
ejde-942	427	33	networks	network	NOUN
ejde-942	427	34	,	,	PUNCT
ejde-942	427	35	132	132	NUM
ejde-942	427	36	(	(	PUNCT
ejde-942	427	37	2020	2020	NUM
ejde-942	427	38	)	)	PUNCT
ejde-942	427	39	,	,	PUNCT
ejde-942	427	40	166	166	NUM
ejde-942	427	41	–	–	SYM
ejde-942	427	42	179	179	NUM
ejde-942	427	43	.	.	PUNCT
ejde-942	428	1	[	[	X
ejde-942	428	2	14	14	NUM
ejde-942	428	3	]	]	X
ejde-942	428	4	g.	g.	PROPN
ejde-942	428	5	karniadakis	karniadakis	PROPN
ejde-942	428	6	,	,	PUNCT
ejde-942	428	7	i.	i.	PROPN
ejde-942	428	8	kevrekidis	kevrekidis	PROPN
ejde-942	428	9	,	,	PUNCT
ejde-942	428	10	l.	l.	PROPN
ejde-942	428	11	lu	lu	PROPN
ejde-942	428	12	,	,	PUNCT
ejde-942	428	13	p.	p.	NOUN
ejde-942	428	14	perdikaris	perdikaris	PROPN
ejde-942	428	15	,	,	PUNCT
ejde-942	428	16	s.	s.	PROPN
ejde-942	428	17	wang	wang	PROPN
ejde-942	428	18	,	,	PUNCT
ejde-942	428	19	l.	l.	PROPN
ejde-942	428	20	yang	yang	PROPN
ejde-942	428	21	;	;	PUNCT
ejde-942	428	22	physics	physics	NOUN
ejde-942	428	23	-	-	PUNCT
ejde-942	428	24	informed	inform	VERB
ejde-942	428	25	machine	machine	NOUN
ejde-942	428	26	learning	learning	NOUN
ejde-942	428	27	,	,	PUNCT
ejde-942	428	28	nature	nature	NOUN
ejde-942	428	29	reviews	review	NOUN
ejde-942	428	30	physics	physics	NOUN
ejde-942	428	31	3	3	NUM
ejde-942	428	32	(	(	PUNCT
ejde-942	428	33	2021	2021	NUM
ejde-942	428	34	)	)	PUNCT
ejde-942	428	35	,	,	PUNCT
ejde-942	428	36	422–440	422–440	NUM
ejde-942	428	37	.	.	PUNCT
ejde-942	429	1	[	[	X
ejde-942	429	2	15	15	NUM
ejde-942	429	3	]	]	X
ejde-942	429	4	l.	l.	PROPN
ejde-942	429	5	lu	lu	PROPN
ejde-942	429	6	,	,	PUNCT
ejde-942	429	7	p.	p.	PROPN
ejde-942	429	8	jin	jin	PROPN
ejde-942	429	9	,	,	PUNCT
ejde-942	429	10	g.	g.	PROPN
ejde-942	429	11	pang	pang	PROPN
ejde-942	429	12	,	,	PUNCT
ejde-942	429	13	z.	z.	PROPN
ejde-942	429	14	zhang	zhang	PROPN
ejde-942	429	15	,	,	PUNCT
ejde-942	429	16	g.	g.	PROPN
ejde-942	429	17	karniadakis	karniadakis	PROPN
ejde-942	429	18	;	;	PUNCT
ejde-942	429	19	learning	learn	VERB
ejde-942	429	20	nonlinear	nonlinear	ADJ
ejde-942	429	21	operators	operator	NOUN
ejde-942	429	22	via	via	ADP
ejde-942	429	23	deeponet	deeponet	NOUN
ejde-942	429	24	based	base	VERB
ejde-942	429	25	on	on	ADP
ejde-942	429	26	the	the	DET
ejde-942	429	27	universal	universal	ADJ
ejde-942	429	28	approximation	approximation	NOUN
ejde-942	429	29	theorem	theorem	NOUN
ejde-942	429	30	of	of	ADP
ejde-942	429	31	operators	operator	NOUN
ejde-942	429	32	,	,	PUNCT
ejde-942	429	33	nature	nature	NOUN
ejde-942	429	34	machine	machine	NOUN
ejde-942	429	35	intelligence	intelligence	NOUN
ejde-942	429	36	,	,	PUNCT
ejde-942	429	37	3	3	NUM
ejde-942	429	38	(	(	PUNCT
ejde-942	429	39	2021	2021	NUM
ejde-942	429	40	)	)	PUNCT
ejde-942	429	41	,	,	PUNCT
ejde-942	429	42	218–229	218–229	NUM
ejde-942	429	43	.	.	PUNCT
ejde-942	430	1	[	[	X
ejde-942	430	2	16	16	NUM
ejde-942	430	3	]	]	PUNCT
ejde-942	430	4	l.	l.	PROPN
ejde-942	430	5	lu	lu	PROPN
ejde-942	430	6	,	,	PUNCT
ejde-942	430	7	x.	x.	PROPN
ejde-942	430	8	meng	meng	PROPN
ejde-942	430	9	,	,	PUNCT
ejde-942	430	10	z.	z.	PROPN
ejde-942	430	11	mao	mao	PROPN
ejde-942	430	12	,	,	PUNCT
ejde-942	430	13	g.	g.	PROPN
ejde-942	430	14	karniadakis	karniadakis	PROPN
ejde-942	430	15	;	;	PUNCT
ejde-942	430	16	deepxde	deepxde	NOUN
ejde-942	430	17	:	:	PUNCT
ejde-942	430	18	a	a	DET
ejde-942	430	19	deep	deep	ADJ
ejde-942	430	20	learning	learning	NOUN
ejde-942	430	21	library	library	NOUN
ejde-942	430	22	for	for	ADP
ejde-942	430	23	solving	solve	VERB
ejde-942	430	24	differential	differential	ADJ
ejde-942	430	25	equations	equation	NOUN
ejde-942	430	26	,	,	PUNCT
ejde-942	430	27	siam	siam	PROPN
ejde-942	430	28	review	review	NOUN
ejde-942	430	29	63	63	NUM
ejde-942	430	30	(	(	PUNCT
ejde-942	430	31	2021	2021	NUM
ejde-942	430	32	)	)	PUNCT
ejde-942	430	33	,	,	PUNCT
ejde-942	430	34	208–228	208–228	NUM
ejde-942	430	35	.	.	PUNCT
ejde-942	431	1	[	[	X
ejde-942	431	2	17	17	NUM
ejde-942	431	3	]	]	PUNCT
ejde-942	431	4	j.	j.	PROPN
ejde-942	431	5	paine	paine	PROPN
ejde-942	431	6	,	,	PUNCT
ejde-942	431	7	r.	r.	PROPN
ejde-942	431	8	anderssen	anderssen	PROPN
ejde-942	431	9	,	,	PUNCT
ejde-942	431	10	f.	f.	PROPN
ejde-942	431	11	de	de	PROPN
ejde-942	431	12	hoog	hoog	PROPN
ejde-942	431	13	;	;	PUNCT
ejde-942	431	14	on	on	ADP
ejde-942	431	15	the	the	DET
ejde-942	431	16	correction	correction	NOUN
ejde-942	431	17	of	of	ADP
ejde-942	431	18	finite	finite	ADJ
ejde-942	431	19	difference	difference	NOUN
ejde-942	431	20	eigenvalue	eigenvalue	NOUN
ejde-942	431	21	approximations	approximation	NOUN
ejde-942	431	22	for	for	ADP
ejde-942	431	23	sturm	sturm	NOUN
ejde-942	431	24	-	-	PUNCT
ejde-942	431	25	liouville	liouville	NOUN
ejde-942	431	26	problems	problem	NOUN
ejde-942	431	27	,	,	PUNCT
ejde-942	431	28	computing	computing	NOUN
ejde-942	431	29	,	,	PUNCT
ejde-942	431	30	26	26	NUM
ejde-942	431	31	(	(	PUNCT
ejde-942	431	32	1981	1981	NUM
ejde-942	431	33	)	)	PUNCT
ejde-942	431	34	,	,	PUNCT
ejde-942	431	35	123–139	123–139	NUM
ejde-942	431	36	.	.	PUNCT
ejde-942	432	1	[	[	X
ejde-942	432	2	18	18	NUM
ejde-942	432	3	]	]	X
ejde-942	432	4	g.	g.	PROPN
ejde-942	432	5	pang	pang	PROPN
ejde-942	432	6	,	,	PUNCT
ejde-942	432	7	l.	l.	PROPN
ejde-942	432	8	lu	lu	PROPN
ejde-942	432	9	,	,	PUNCT
ejde-942	432	10	g.	g.	PROPN
ejde-942	432	11	karniadakis	karniadakis	PROPN
ejde-942	432	12	;	;	PUNCT
ejde-942	432	13	fpinns	fpinns	ADJ
ejde-942	432	14	:	:	PUNCT
ejde-942	432	15	fractional	fractional	ADJ
ejde-942	432	16	physics	physics	NOUN
ejde-942	432	17	-	-	PUNCT
ejde-942	432	18	informed	inform	VERB
ejde-942	432	19	neural	neural	ADJ
ejde-942	432	20	networks	network	NOUN
ejde-942	432	21	,	,	PUNCT
ejde-942	432	22	siam	siam	ADJ
ejde-942	432	23	journal	journal	NOUN
ejde-942	432	24	on	on	ADP
ejde-942	432	25	scientific	scientific	ADJ
ejde-942	432	26	computing	computing	NOUN
ejde-942	432	27	,	,	PUNCT
ejde-942	432	28	41	41	NUM
ejde-942	432	29	(	(	PUNCT
ejde-942	432	30	2019	2019	NUM
ejde-942	432	31	)	)	PUNCT
ejde-942	432	32	,	,	PUNCT
ejde-942	432	33	a2603	a2603	PROPN
ejde-942	432	34	–	–	PUNCT
ejde-942	432	35	a2626	a2626	NOUN
ejde-942	432	36	.	.	PUNCT
ejde-942	433	1	[	[	X
ejde-942	433	2	19	19	NUM
ejde-942	433	3	]	]	PUNCT
ejde-942	433	4	m.	m.	NOUN
ejde-942	433	5	raissi	raissi	NOUN
ejde-942	433	6	,	,	PUNCT
ejde-942	433	7	p.	p.	NOUN
ejde-942	433	8	perdikaris	perdikaris	NOUN
ejde-942	433	9	,	,	PUNCT
ejde-942	433	10	g.	g.	PROPN
ejde-942	433	11	karniadakis	karniadakis	PROPN
ejde-942	433	12	;	;	PUNCT
ejde-942	433	13	physics	physics	NOUN
ejde-942	433	14	-	-	PUNCT
ejde-942	433	15	informed	inform	VERB
ejde-942	433	16	neural	neural	ADJ
ejde-942	433	17	networks	network	NOUN
ejde-942	433	18	:	:	PUNCT
ejde-942	433	19	a	a	DET
ejde-942	433	20	deep	deep	ADJ
ejde-942	433	21	learning	learning	NOUN
ejde-942	433	22	framework	framework	NOUN
ejde-942	433	23	for	for	ADP
ejde-942	433	24	solving	solve	VERB
ejde-942	433	25	forward	forward	ADV
ejde-942	433	26	and	and	CCONJ
ejde-942	433	27	inverse	inverse	NOUN
ejde-942	433	28	problems	problem	NOUN
ejde-942	433	29	involving	involve	VERB
ejde-942	433	30	nonlinear	nonlinear	ADJ
ejde-942	433	31	partial	partial	ADJ
ejde-942	433	32	differential	differential	NOUN
ejde-942	433	33	equations	equation	NOUN
ejde-942	433	34	,	,	PUNCT
ejde-942	433	35	journal	journal	NOUN
ejde-942	433	36	of	of	ADP
ejde-942	433	37	computational	computational	ADJ
ejde-942	433	38	physics	physics	NOUN
ejde-942	433	39	,	,	PUNCT
ejde-942	433	40	378	378	NUM
ejde-942	433	41	(	(	PUNCT
ejde-942	433	42	2019	2019	NUM
ejde-942	433	43	)	)	PUNCT
ejde-942	433	44	,	,	PUNCT
ejde-942	433	45	686–707	686–707	NUM
ejde-942	433	46	.	.	PUNCT
ejde-942	434	1	[	[	X
ejde-942	434	2	20	20	NUM
ejde-942	434	3	]	]	PUNCT
ejde-942	434	4	l.	l.	PROPN
ejde-942	434	5	yang	yang	PROPN
ejde-942	434	6	,	,	PUNCT
ejde-942	434	7	d.	d.	PROPN
ejde-942	434	8	zhang	zhang	PROPN
ejde-942	434	9	,	,	PUNCT
ejde-942	434	10	g.	g.	PROPN
ejde-942	434	11	karniadakis	karniadakis	PROPN
ejde-942	434	12	;	;	PUNCT
ejde-942	434	13	physics	physics	NOUN
ejde-942	434	14	-	-	PUNCT
ejde-942	434	15	informed	inform	VERB
ejde-942	434	16	generative	generative	ADJ
ejde-942	434	17	adversarial	adversarial	ADJ
ejde-942	434	18	networks	network	NOUN
ejde-942	434	19	for	for	ADP
ejde-942	434	20	stochastic	stochastic	ADJ
ejde-942	434	21	differential	differential	ADJ
ejde-942	434	22	equations	equation	NOUN
ejde-942	434	23	,	,	PUNCT
ejde-942	434	24	siam	siam	ADJ
ejde-942	434	25	journal	journal	NOUN
ejde-942	434	26	on	on	ADP
ejde-942	434	27	scientific	scientific	ADJ
ejde-942	434	28	computing	computing	NOUN
ejde-942	434	29	,	,	PUNCT
ejde-942	434	30	42	42	NUM
ejde-942	434	31	(	(	PUNCT
ejde-942	434	32	2020	2020	NUM
ejde-942	434	33	)	)	PUNCT
ejde-942	434	34	,	,	PUNCT
ejde-942	434	35	a292	a292	PROPN
ejde-942	434	36	–	–	PUNCT
ejde-942	434	37	a317	a317	NUM
ejde-942	434	38	.	.	PUNCT
ejde-942	435	1	[	[	X
ejde-942	435	2	21	21	NUM
ejde-942	435	3	]	]	X
ejde-942	435	4	y.	y.	PROPN
ejde-942	435	5	zang	zang	PROPN
ejde-942	435	6	,	,	PUNCT
ejde-942	435	7	g.	g.	PROPN
ejde-942	435	8	bao	bao	PROPN
ejde-942	435	9	,	,	PUNCT
ejde-942	435	10	x.	x.	NOUN
ejde-942	435	11	ye	ye	PROPN
ejde-942	435	12	,	,	PUNCT
ejde-942	435	13	h.	h.	PROPN
ejde-942	435	14	zhou	zhou	PROPN
ejde-942	435	15	;	;	PUNCT
ejde-942	435	16	weak	weak	ADJ
ejde-942	435	17	adversarial	adversarial	ADJ
ejde-942	435	18	networks	network	NOUN
ejde-942	435	19	for	for	ADP
ejde-942	435	20	high	high	ADJ
ejde-942	435	21	-	-	PUNCT
ejde-942	435	22	dimensional	dimensional	ADJ
ejde-942	435	23	partial	partial	ADJ
ejde-942	435	24	differential	differential	NOUN
ejde-942	435	25	equations	equation	NOUN
ejde-942	435	26	,	,	PUNCT
ejde-942	435	27	journal	journal	NOUN
ejde-942	435	28	of	of	ADP
ejde-942	435	29	computational	computational	ADJ
ejde-942	435	30	physics	physics	NOUN
ejde-942	435	31	,	,	PUNCT
ejde-942	435	32	411	411	NUM
ejde-942	435	33	(	(	PUNCT
ejde-942	435	34	2020	2020	NUM
ejde-942	435	35	)	)	PUNCT
ejde-942	435	36	,	,	PUNCT
ejde-942	435	37	109409	109409	NUM
ejde-942	435	38	.	.	PUNCT
ejde-942	436	1	[	[	X
ejde-942	436	2	22	22	NUM
ejde-942	436	3	]	]	X
ejde-942	436	4	j.	j.	PROPN
ejde-942	436	5	zu	zu	PROPN
ejde-942	436	6	;	;	PUNCT
ejde-942	436	7	approximate	approximate	ADJ
ejde-942	436	8	stabilization	stabilization	NOUN
ejde-942	436	9	of	of	ADP
ejde-942	436	10	one	one	NUM
ejde-942	436	11	-	-	PUNCT
ejde-942	436	12	dimensional	dimensional	ADJ
ejde-942	436	13	schrödinger	schrödinger	NOUN
ejde-942	436	14	equations	equation	NOUN
ejde-942	436	15	in	in	ADP
ejde-942	436	16	inhomogeneous	inhomogeneous	ADJ
ejde-942	436	17	media	medium	NOUN
ejde-942	436	18	,	,	PUNCT
ejde-942	436	19	journal	journal	NOUN
ejde-942	436	20	of	of	ADP
ejde-942	436	21	optimization	optimization	NOUN
ejde-942	436	22	theory	theory	NOUN
ejde-942	436	23	and	and	CCONJ
ejde-942	436	24	applications	application	NOUN
ejde-942	436	25	,	,	PUNCT
ejde-942	436	26	153	153	NUM
ejde-942	436	27	(	(	PUNCT
ejde-942	436	28	2012	2012	NUM
ejde-942	436	29	)	)	PUNCT
ejde-942	436	30	,	,	PUNCT
ejde-942	436	31	758–768	758–768	NUM
ejde-942	436	32	.	.	PUNCT
ejde-942	437	1	sen	sen	PROPN
ejde-942	437	2	zhang	zhang	PROPN
ejde-942	437	3	center	center	PROPN
ejde-942	437	4	for	for	ADP
ejde-942	437	5	mathematics	mathematic	NOUN
ejde-942	437	6	and	and	CCONJ
ejde-942	437	7	interdisciplinary	interdisciplinary	ADJ
ejde-942	437	8	sciences	science	NOUN
ejde-942	437	9	,	,	PUNCT
ejde-942	437	10	and	and	CCONJ
ejde-942	437	11	school	school	NOUN
ejde-942	437	12	of	of	ADP
ejde-942	437	13	mathematics	mathematic	NOUN
ejde-942	437	14	and	and	CCONJ
ejde-942	437	15	statistics	statistic	NOUN
ejde-942	437	16	,	,	PUNCT
ejde-942	437	17	northeast	northeast	ADJ
ejde-942	437	18	normal	normal	ADJ
ejde-942	437	19	university	university	NOUN
ejde-942	437	20	,	,	PUNCT
ejde-942	437	21	changchun	changchun	PROPN
ejde-942	437	22	,	,	PUNCT
ejde-942	437	23	130024	130024	NUM
ejde-942	437	24	,	,	PUNCT
ejde-942	437	25	china	china	PROPN
ejde-942	437	26	email	email	NOUN
ejde-942	437	27	address	address	NOUN
ejde-942	437	28	:	:	PUNCT
ejde-942	437	29	zhangs832@nenu.edu.cn	zhangs832@nenu.edu.cn	X
ejde-942	437	30	jian	jian	PROPN
ejde-942	437	31	zu	zu	PROPN
ejde-942	437	32	(	(	PUNCT
ejde-942	437	33	corresponding	corresponding	ADJ
ejde-942	437	34	author	author	NOUN
ejde-942	437	35	)	)	PUNCT
ejde-942	437	36	center	center	NOUN
ejde-942	437	37	for	for	ADP
ejde-942	437	38	mathematics	mathematic	NOUN
ejde-942	437	39	and	and	CCONJ
ejde-942	437	40	interdisciplinary	interdisciplinary	ADJ
ejde-942	437	41	sciences	science	NOUN
ejde-942	437	42	,	,	PUNCT
ejde-942	437	43	and	and	CCONJ
ejde-942	437	44	school	school	NOUN
ejde-942	437	45	of	of	ADP
ejde-942	437	46	mathematics	mathematic	NOUN
ejde-942	437	47	and	and	CCONJ
ejde-942	437	48	statistics	statistic	NOUN
ejde-942	437	49	,	,	PUNCT
ejde-942	437	50	northeast	northeast	ADJ
ejde-942	437	51	normal	normal	ADJ
ejde-942	437	52	university	university	NOUN
ejde-942	437	53	,	,	PUNCT
ejde-942	437	54	changchun	changchun	PROPN
ejde-942	437	55	,	,	PUNCT
ejde-942	437	56	130024	130024	NUM
ejde-942	437	57	,	,	PUNCT
ejde-942	437	58	china	china	PROPN
ejde-942	437	59	email	email	NOUN
ejde-942	437	60	address	address	NOUN
ejde-942	437	61	:	:	PUNCT
ejde-942	437	62	zuj100@nenu.edu.cn	zuj100@nenu.edu.cn	NUM
ejde-942	437	63	jingqi	jingqi	NOUN
ejde-942	437	64	zhang	zhang	PROPN
ejde-942	437	65	center	center	PROPN
ejde-942	437	66	for	for	ADP
ejde-942	437	67	mathematics	mathematic	NOUN
ejde-942	437	68	and	and	CCONJ
ejde-942	437	69	interdisciplinary	interdisciplinary	ADJ
ejde-942	437	70	sciences	science	NOUN
ejde-942	437	71	,	,	PUNCT
ejde-942	437	72	and	and	CCONJ
ejde-942	437	73	school	school	NOUN
ejde-942	437	74	of	of	ADP
ejde-942	437	75	mathematics	mathematic	NOUN
ejde-942	437	76	and	and	CCONJ
ejde-942	437	77	statistics	statistic	NOUN
ejde-942	437	78	,	,	PUNCT
ejde-942	437	79	northeast	northeast	ADJ
ejde-942	437	80	normal	normal	ADJ
ejde-942	437	81	university	university	NOUN
ejde-942	437	82	,	,	PUNCT
ejde-942	437	83	changchun	changchun	PROPN
ejde-942	437	84	,	,	PUNCT
ejde-942	437	85	130024	130024	NUM
ejde-942	437	86	,	,	PUNCT
ejde-942	437	87	china	china	PROPN
ejde-942	437	88	email	email	NOUN
ejde-942	437	89	address	address	NOUN
ejde-942	437	90	:	:	PUNCT
ejde-942	437	91	zhangjq906@nenu.edu.cn	zhangjq906@nenu.edu.cn	X
ejde-942	437	92	1	1	X
ejde-942	437	93	.	.	PUNCT
ejde-942	438	1	introduction	introduction	NOUN
ejde-942	438	2	2	2	NUM
ejde-942	438	3	.	.	PUNCT
ejde-942	438	4	model	model	PROPN
ejde-942	438	5	problem	problem	NOUN
ejde-942	438	6	3	3	X
ejde-942	438	7	.	.	PUNCT
ejde-942	438	8	quantitative	quantitative	ADJ
ejde-942	438	9	estimation	estimation	NOUN
ejde-942	438	10	of	of	ADP
ejde-942	438	11	eigenpairs	eigenpair	NOUN
ejde-942	438	12	4	4	NUM
ejde-942	438	13	.	.	NOUN
ejde-942	438	14	comparison	comparison	NOUN
ejde-942	438	15	with	with	ADP
ejde-942	438	16	previous	previous	ADJ
ejde-942	438	17	algorithms	algorithm	NOUN
ejde-942	438	18	4.1	4.1	NUM
ejde-942	438	19	.	.	PUNCT
ejde-942	439	1	comparison	comparison	NOUN
ejde-942	439	2	with	with	ADP
ejde-942	439	3	the	the	DET
ejde-942	439	4	state	state	NOUN
ejde-942	439	5	-	-	PUNCT
ejde-942	439	6	of	of	ADP
ejde-942	439	7	-	-	PUNCT
ejde-942	439	8	the	the	DET
ejde-942	439	9	-	-	PUNCT
ejde-942	439	10	art	art	NOUN
ejde-942	439	11	deep	deep	ADJ
ejde-942	439	12	learning	learning	NOUN
ejde-942	439	13	methods	method	NOUN
ejde-942	439	14	4.2	4.2	NUM
ejde-942	439	15	.	.	PUNCT
ejde-942	440	1	comparison	comparison	NOUN
ejde-942	440	2	with	with	ADP
ejde-942	440	3	classical	classical	ADJ
ejde-942	440	4	numerical	numerical	ADJ
ejde-942	440	5	methods	method	NOUN
ejde-942	440	6	5	5	NUM
ejde-942	440	7	.	.	PUNCT
ejde-942	440	8	numerical	numerical	ADJ
ejde-942	440	9	experiments	experiment	NOUN
ejde-942	440	10	6	6	NUM
ejde-942	440	11	.	.	PUNCT
ejde-942	441	1	conclusion	conclusion	NOUN
ejde-942	441	2	references	reference	NOUN