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ETASR - Engineering, Technology & Applied Science Research Vol. 2, �o. 1, 2012, 167-172 167  
  

www.etasr.com Zjavka: Recognition of Generalized Patterns by a Differential Polynomial �eural �etwork 

 

Recognition of Generalized Patterns by a Differential 
Polynomial Neural Network 

 

Ladislav Zjavka 
Faculty of Management Science and Informatics  

University of Žilina 
Žilina, Slovakia 

lzjavka@gmail.com 
 
 

Abstract — A lot of problems involve unknown data relations, 

identification of which can serve as a generalization of their 

qualities. Relative values of variables are applied in this case, and 

not the absolute values, which can better make use of data 

properties in a wide range of the validity. This resembles more to 

the functionality of the brain, which seems to generalize relations 

of variables too, than a common pattern classification. 

Differential polynomial neural network is a new type of neural 

network designed by the author, which constructs and 

approximates an unknown differential equation of dependent 

variables using special type of root multi-parametric 

polynomials. It creates fractional partial differential terms, 

describing mutual derivative changes of some variables, likewise 

the differential equation does. Particular polynomials catch 

relations of given combinations of input variables. This type of 

identification is not based on a whole-pattern similarity, but only 

to the learned hidden generalized relations of variables. 

Keywords - polynomial neural network; dependence of 

variables identification; differential equation approximation; 

rational integral function 

I.  INTRODUCTION 

The principal disadvantage of the artificial neural network 
(ANN) identification in general is the disability of input pattern 
generalization. ANNs can learn to classify any input patterns 
but utilize only the absolute values of variables. However, the 
latter may differ significantly while their relations may be the 
same. That is why ANNs are able to correctly recognize only 
similar or incomplete patterns compared to the train set. If the 
input considered is e.g. a shape moved or sized in the input 
matrix, the neural network identification will fail. An approach 
to look at the input vector of variables not as a “pattern” but as 
a dependent bound point set of N-dimensional space could be 
attempted. A neural network, which would be able to learn and 
identify any unknown data relations, is to contain a multi-
parametric polynomial functions to catch partial dependence of 
given inputs. Its response would be the same to all patterns 
(sets) which variables are performed with the trained 
dependence, regardless of the actual values [9]. Biological 
neural cell seems to apply a similar principle. Its dendrites 
collect electrical signals coming from other neurons. But unlike 
the artificial neuron, some of the signals already interact in 
single branches (dendrites) of a neural cell (see Figure 1), 

likewise the multiplied variables of a multi-parametric 
polynomial do.  

Parameters of polynomial terms can represent the synopsis 
of the cell dendrites. These weighted combinations are summed 
in the body cell and transformed into relative values using 
time-delayed dynamic periodic activation functions (the 
activated neural cell generates series of time-delayed output 
pulses, in response to its input signals). Axon passes electrical 
pulse signals on to dendrites of other neural or effector cells 
[1]. The period of this function depends on some input 
variables and seems to represent the derivative part of a partial 
term of a differential equation composition. Differential 
polynomial neural network (D-PNN) constructs and tries to 
approximate an unknown differential equation describing 
relations of input variables that are not entirely patterns. It 
forms its output as a generalization of input patterns similar to 
the ones utilized by the human brain. It creates a structural 
model of any unknown relationships of input variables 
description. D-PNN is based on GMDH (Group Method of 
Data Handling) polynomial neural network, which was created 
by the Ukrainian scientist Aleksey Ivakhnenko in 1968, when 
the back-propagation technique was not known yet. He 
attempted to decompose the complexity of a process into many 
simpler relationships each described by a low order 2-variable 
polynomial processing function of a single neuron [2].  

 

 

Fig. 1.   A biological neural cell 



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II. DIFFERENTIAL POLYNOMIAL NEURAL NETWORK 

The basic idea of the D-PNN is to create and approximate a 
differential equation (DE) (3), which is not known in advance 
[3], with a special type of root (power) fractional multi-
parametric polynomials (5). 

 

∑∑∑∑
∞

====

==+
∂∂

∂
+

∂
∂

+
11

2

11

0...
k

k

n

j ji

ij

n

i

n

i i

i
uu

xx

u
c

x

u
ba

          (3) 

u(x1, x2,, … , xn) - searched  function of all input variables  

a, B(b1, b2,, ..., bn), C(c11, c12, ,... ) - parameters 
 
    Fourier’s method of partial DE solution searches the 
solution in a form of the product of 2 functions, of which at 
least 1 depends only on 1 variable. A partial derivation of 
function z(x, y) of 2 input variables x, y can be expressed by 
(4) [4]. 

)()(
),(

21 zfxf
x

yxz
⋅=

∂

∂    (4) 

 
    Elementary methods of a differential equation solution 
express the solution in special elementary functions – 
polynomials (e.g. Bessel’s functions, Fourier’s power series). 
Numerical integration of differential equations is based on 
their approximation through: 

• rational integral functions  
• trigonometric series 

    The 1st, and more simple way, has been selected, using 
the method of integral analogues, which replaces mathematical 
operators and symbols in DE by the ratio of corresponding 
variables. Derivatives are replaced by the integral analogues, 
i.e. derivative operators are removed and simultaneously all 
operators are replaced by similarly or proportion marks in 
equations, all vectors are replaced by their absolute values. 
Dimensional terms are divided by some others, which results in 
searched non-dimensional likeness criterions [5]. 

( )
( ) m

n

m

m

n
nnn

i
xxx

xxxf

xbb

xxaxaxaxaa
y

∂∂∂

∂
=

++

++++++
= +

...

),...,,(

...

......

21

21

1

110

1

21122110
  (5) 

n – combination degree of  n-input variable polynomial of numerator 
m – combination degree of denominator (m<n) 

 

0...
......

.........
..

...

......

110

0

110

210
10 =+

++++

+++++
++

++
+++

+=
vur

kjiqjip

t
k

xxbxbb

xxxaxxaa
w

xbb

xxaa
wwY

(6) 

 wt – weights of  terms  

 
    The fractional polynomials (5), which can describe a partial 
dependence of n-input variables of each neuron, are applied as 
terms of the DE (6) composition. They partly create an 
unknown multi-parametric non-linear function, which codes 
relations of input variables. The numerator of (5) is a 
polynomial of complete n-input combination degree of a single 
neuron and realizes a new function z of formula (4). The 
denominator of (5) is a derivative part, which gives a partial 
mutual change of some neuron input variables and its 
polynomial combination degree m is less then n. It arose from 
the partial derivation of the complete n-variable polynomial by 
competent variable(s). 

 

Fig. 2.  A block of differential neurons 

    Each layer of the D-PNN consists of blocks, which contain 
derivative neurons, one for each fractional polynomial (5), 
defining the partial derivative dependent change of some input 
variables. A block also contains additional extended neurons 
(EN), which form compound functions (15) applying previous 
layer block outputs. Each block contains a single polynomial 
(without derivative part), which forms its output entrance into 
the next hidden layer (Figure 2.). Neurons don’t affect the 
block output but are applied only for the total output 
calculation (DE composition). Each neuron has 2 vectors of 
adjustable parameters a, b and each block contains 1 vector of 
adjustable parameters of the output polynomial. The root 
functions of denominators (5) are lower than n, according to 
the combination degree, which take the polynomials of neurons 
into competent power degree. They can be replaced by power 
functions of denominators. Inputs of constant combination 
degree (n=2,3,…) forming particular combinations of variables, 
enter each block, where they are substituted into polynomials 
(Figure 3.). It is necessary to adjust not only the polynomial 
parameters, but also the D-PNN‘s structure. This means some 
neurons in terms of role of the DE are to be left out. 

 

 

Fig. 3.  Differential polynomial neural network 

x1             x2              x3 

Block output 

polynomial 

�   neurons (5) 

Combination 
degree n=3 

Extended 

neurons 

 /  / 

Π 



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III. IDENTIFICATION OF SIMPLE LINEAR DEPENDENCIES 

Consider a very simple dependence of 2-input variables, 
which multiplicity is constant (e.g. =2). D-PNN will contain 
only 1 block of 2 polynomial neurons (7)(8) as terms of DE 
(Figure 4.). As the input variables don’t change constantly, it is 
necessary to add both terms (fractional polynomial of 
derivative variable x1 and x2) in the DE (block). D-PNN will 
learn this relation easily according to samples of the training 
data set by means of genetic and evolution algorithm (GA) [7]. 

( )
110

2
1

21322110
11

xbb

xxaxaxaa
wy

+
+++

=
  (7) 

( )
210

2
1

21322110
22

xbb

xxaxaxaa
wy

+

+++
=

  (8) 

 
Consider a more complicated linear dependence, where 2 

variables depend on a 3rd. For example sum of the first 2 
variables equals the 3rd variable (x1 + x2 = x3). The complete 
DE (for derivatives 1 and 2-combinations of block) consists of 
6 terms (neurons) but only 3 of them will be enough for 
derivative terms x3 (9), x1x3 (10), x2x3 (11). If other terms 
(neurons) are added, the D-PNN will work amiss (see Figure 
5). Two-variable combination polynomials of numerators 
(7)(8) can be also applied, which could improve the D-PNN 
functionality and increase the number of the DE terms. This 3-
variable dependence is described by more complicated 
exponential functions. The D-PNN as well is charged by the 
possible 2-sided dependent change of input variables. For 
example 1+9=10 is the same sum as 9+1=10. The principal 
phase of its adjustment resides in eliminating of some neurons 
(in terms of the DE). 

 

 

Fig. 4.  Identification of a constant quotient of 2 variables (x1 = 2x2 ) 

 

 

Fig. 5.  Identification of the sum dependence 

( )
310

3
1

32172143322110
11

...

xbb

xxxaxxaxaxaxaa
wy

+
++++++

=
       (9) 

( )
( ) 2131332110

3
1

32172143322110
22

...

xxbxbxbb

xxxaxxaxaxaxaa
wy

+++

++++++
=

       (10) 

( )
( ) 2132332210

3
1

32172143322110
33

...

xxbxbxbb

xxxaxxaxaxaxaa
wy

+++

++++++
=

       (11) 

 
Multi-layered D-PNN creates compound polynomial 

functions. Main exponential functions of higher layers “carry” 
some secondary functions of previous layers, describing the 
partial relations of its variables. From mathematical point of 
view the 1st hidden layer forms the inner functions, which 
substitute the input variables of 2nd hidden layer neuron and 
block polynomials - the outer functions. Provided this 
assumption we are able to calculate the partial derivatives of 
compound functions by variables of previous layers as DE 
terms (14), from the inner functions (12) of an outer function 
(13). These compound DE terms are formed as products of 
partial derivatives of main and inner functions (15) [6]. 

),...,,()( 21 niii xxxXy ϕϕ ===  i=1, … , m   (12) 

F(x1, x2, … , xn) = f(y1, y2, … , ym) = f(φ1(X), φ2(X),..., φm(X))         (13) 
















∂

∂
⋅

∂
∂

++
∂

∂
⋅

∂
∂

+
∂

∂
⋅

∂
∂

=
∂
∂

∂

∂
⋅

∂
∂

++
∂

∂
⋅

∂
∂

+
∂

∂
⋅

∂
∂

=
∂
∂

∂

∂
⋅

∂
∂

++
∂

∂
⋅

∂
∂

+
∂

∂
⋅

∂
∂

=
∂
∂

n

m

mnnn

m

m

m

m

xy

f

xy

f

xy

f

x

F

xy

f

xy

f

xy

f

x

F

xy

f

xy

f

xy

f

x

F

ϕϕϕ

ϕϕϕ

ϕϕϕ

...

..........

...

...

2

2

1

1

22

2

22

1

12

11

2

21

1

11

  (14) 

∑
= ∂

∂
⋅

∂

∂
=

∂
∂ m

i k

i

i

m

k
x

X

y

yyyf

x

F

1

21 )(),...,,( φ  k=1, … , n (15) 

 

 
 

Fig. 6.  Identification of the 3-variable dependence with 2-combination 
blocks 

      x1               /                  x2      =       2 

Block output 

 w0    +                          +                                                                                                                                 
 

         Neuron 1                    Neuron 2 

 

 (7) (8) 

    x1        +         x2         =         x3 

Block output 

 
 

 w0   +                    +                    + 
 

      Neuron 1               Neuron 2                Neuron 3 

(9) (10) (11) 



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Each block of the D-PNN forms partial DE terms utilizing 
its basic and extended neurons. Single adjustable polynomial (P 
in Figure 6.) without derivative part creates the block output 
(applying in the next hidden layer) but the neurons are applied 
only for the total DE composition. The blocks of the 2nd and 
the following hidden layers create compound terms (CT) of the 
DE using their additional extended neurons, outputs and inputs 
of back connected blocks of previous layers. Consider for 
instance the 1st block of the last hidden layer, which takes its 
own neurons as 2 basic terms (16) of the DE (6). Subsequently 
it creates 4 extended terms of the 2nd (previous) hidden layer 
variables, using reverse output polynomials and inputs of 2 
bound blocks. It creates 4 fractional compound terms of the DE 
for 4 derivative input variables of previous hidden layer using 
derivations of compound and inner functions (17). As couples 
of variables of the inner functions φ1 (x1, x2) and φ1 (x3, x4) 
differ from each other, their partial derivations are = 0 and so 
the sum (15) will consist only of 1 term.  

( )
( )

( )
( ),,110

1
1

1,,
110

,,
2

,,
13

,,
22

,,
110

1
1
1

22

2
1

2
1

xbb

x
w

xbb

xxaxaxaa
wy

+⋅
=

+⋅

++
=

  (16) 

( ),110
,,

1

,,
2

,,,
1

3

3
1
3

2

)()( 2
1

2
1

xcc

x

x

x
wy

+⋅
⋅=

    (17) 

( )110

,
1

,
2

,,
1

,,
2

,,,
1

7

7
1
7

2

)()()( 2
1

2
1

2
1

xdd

x

x

x

x

x
wy

+⋅
⋅⋅=    (18) 

 
The previous layer block reverse outputs are used to create 

necessary partial derivations of the outer and inner functions 
(of polynomials) of differential neurons (17). Likewise 
compound terms can be created for the 1st hidden layer (18). 
The 3 linked blocks forming 8 terms of the DE were attached 
to the presently adjusted block. This can be performed well by 
a recursive algorithm. It was not every term that was used in 
the complete DE; some of them were necessarily left out. This 
indicates “0” or “1” in the neurons of blocks and is ease to use 
them as genes of GA. Parameters of polynomials are 
represented by real numbers. A chromosome is a sequence of 
their values, which can be easy mutated. The D-PNN’s total 
output Y is the sum of all active partial DE term values 
according to (19), which the present active amount a can be 
built in. 

a

y

Y

k

i

i∑
== 1   k = total amount of DE terms (19) 

 

It can be seen, that the 3-variable D-PNN (Figure 6.) 
substantially consists of 3 overlaying “wedge” networks (WN), 
each going back out from the blocks of the last hidden layer 
and gradually attaching to the derivative variables of previous 
layers. The D-PNN of the 4 dependent input variables using 2-
combination blocks will have totally 6 blocks of all input 
combination couples in the 1st hidden layer. The number of 
combinations for all variables increases enormously each next 
hidden layer. This could be solved by applying WNs, as only 
some of the blocks are created and used. The total amount of 
D-PNN’s hidden layers could equal at least to the number of 
input variables (i.e. 4), as it must be able to create each 
combination of which and to reach back all derivative variables 

of the 1st layer. So WNs of the 1st hidden layer will involve 
min. 4 random blocks, consequently in the 2nd layer will 
contain min. 3 blocks, etc. This way the number of all WN 
blocks decreases each next hidden layer until is reached just 1 
block. D-PNN will have several overlaying WNs partly in the 
layers again. Some WN layers overlay each other and so the 
blocks can be used several times by different WNs (Figure 7.). 
The blocks of the 2nd and following hidden layers can be 
reconnected and this could compensate missing combination 
blocks. The connections of the complete 1st hidden layer blocks 
are fixed. Likewise the previous 3-variable D-PNN type does, 
it can construct the partial fractional terms of the DE from 
back-connected blocks of previous layers. All WN blocks 
attach back gradually the derivative variables of previous 
layers. The searching space contains a great amount of local 
error solutions, which GA can finish easily. This problem is 
caused by a lot of possible combinations of block inputs and 
composed DE terms (only some of them may be employed), 
which selection is a critical phase of the D-PNN’s construction, 
besides the simultaneous parameter adjustment [10]. 

 

 

Fig. 7.  “Wedge” networks of the 4-variable 2-combination D-PNN 

The D-PNN of the 6 dependent input variables using 2-
combination blocks will have totally 15 blocks of all input 
combination couples in the 1st hidden layer and 6 hidden layers. 
However in experiment with right triangles (Figure 9 and 
Figure 10.) it could be sufficient with 4 hidden layers again, 
because there is the maximum of 4-variable dependence to 
identify. 

IV. EXPERIMENTS 

4-variable D-PNN is able to identify row/column or 
diagonal dependence of chess pieces (Figure 8.). Input vector is 
formed by their x, y positions (row, column). If the white rook 
checks the black bishop their x or y positions equal and this can 
D-PNN learn to identify. Another relation occurs if the black 
bishop checks the white rook, the sum or difference of their x 
and y-positions are equal Ax+Ay=Bx+By or Ax–Ay=Bx–By. Table 
1 and Table 2 show network responses to dependent and 

      

  x1        +          x2        =           x3         +         x4 

     
� 

    

  

� 

� 

DE 

 

�.. 



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independent variables of input vectors. Training data set can 
consist of following 5 data samples x and y-positions of the 
chess pieces (the chessboard was enlarged) [9]:    

{1+22=20+3}, {16+1=2+15}, {34+3=30+7}, {60+30=42+48}, 
{5+25=2+28}, {2+5=4+3} 

 
              1           2          3          4          5          6 

 

 

 

 

 

 
 

Fig. 8.  Relations of chess pieces 

TABLE I.  RESPONSES TO RANDOM INPUT VECTORS WITH 4 DEPENDENT 
VARIABLES 

Input vector Output 

31 + 46 = 11 + 66 0.9654 
18 + 57 = 24 + 51 1.0273 
46 + 30 = 25 + 51 1.0018 
58 + 44 = 17 + 85 1.0006 
14 + 25 = 1 + 38 0.9882 
4 + 23 = 8 + 19 1.0031 

TABLE II.  RESPONSES TO RANDOM INDEPENDENT INPUT VECTORS OF 4 
VARIABLES 

Input vector Output Input vector Output 

46 + 55 = 3 + 108 (-10) 1.2041 25 + 45 = 51 + 9 (+10) 0.8461 
11 + 55 < 27 + 49 1.1865 57 + 54 > 43 + 58 0.9564 
56 + 49 < 5 + 110 1.1213 15 + 42 > 12 + 35 0.9179 
8 + 29 < 21 + 26 1.1587 49 + 47 > 52 + 34 0.9162 
45 + 34 < 2 + 87 1.1917 31 + 32 > 37 + 16 0.8704 
3 + 39 < 26 + 26 1.4803 45 + 56 > 30 + 61 0.9515 
35 + 9 < 18 + 36 1.2038 27 + 58 > 45 + 30 0.9228 
29 + 6 < 37 + 8 1.1653 44 + 34 > 36 + 32 0.9267 
60 + 43 < 41 + 72 1.0526 39 + 25 > 30 + 24 0.9174 
5 + 40 < 40 + 15 1.2457 59 + 39 > 51 + 37 0.9414 

 

A separating plane could be noticed, detached from the 
relative “classes”, which have the same characteristic (if the 
sum of the 1st couple is less then it should be, the output is less 
then the desired round and other hand round). D-PNN can be 
trained only with small input-output data samples (likewise the 
GMDH polynomial neural network does) to learn any 
dependence [8]. 

6-variable D-PNN can learn to identify (generalize) a 
changeable shape (e.g. triangle) regardless of its size or 

position in the input matrix (Figure 9.). Input vector of the D-
PNN is formed by x, y (row, column) coordinates of the 3 
triangle apexes A, B, C. The dependence of points A, C is 
diagonal, A, B vertical and B, C horizontal. As there are 
simultaneously occurred 3 types of point relations, it is 
necessary to increase the number of blocks of D-PNN’s hidden 
layers. Training data set can consist of following 5 data 
samples – right equal-based triangles : 

{6,44,11,44,11,49}, {22,13,32,13,32,23}, {10,30,25,30,25,45}, 
{3,20,23,20,23,40}, {25,50,50,50,50,75} 

Testing random right triangles must keep the apexes A, B, C 
dependent to be correctly recognized by D-PNN. Table 3 
shows responses of trained network to dependent right 
triangles. Table 4. applies only vertical deformations (+ and –) 
of right triangles in C apexes (Fig.10. down), to be shown a 
transparent separating plane, detaching the relative triangles. 

 

 

Fig. 9.  Dependent right equal-sided triangle shapes 

 

 

Fig. 10.  Independent deformed right triangles in apexes A and C 

1 

2 

3 

4 

5 

6 



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TABLE III.  RESPONSES TO RANDOM INPUT VECTORS WITH 6 DEPENDENT 
VARIABLES (RIGHT TRIANGLES) 

Input vector (Ax,Ay Bx,By Cx,Cy) Output 

34,44 60,44 60,70 1.0142 
18,5 35,5 35,22 0.9692 
15,39 35,39 35,59 0.9932 
16,51 58,51 58,93 1.0095 
29,51 45,51 45,67 1.0227 
21,5 29,5 29,13 1.067 

TABLE IV.  RESPONSES TO RANDOM INDEPENDENT INPUT VECTORS OF 6 
VARIABLES (DEFORMED  RIGHT TRIANGLES IN C APEXES) 

Input vector (A B C) Output Input vector (A B C) Output 

13,36 42,36 33,65 0.8750 24,15 34,15 37,25 1.0713 
21,27 59,27 47,65 0.8803 4,11 40,11 52,47 1.2376 
9,10 34,10 26,35 0.8605 48,40 56,40 58,48 1.1255 
12,43 25,43 21,56 0.9096 32,22 57,22 65,47 1.0885 
35,10 57,10 50,32 0.9523 18,28 25,28 27,35 1.0558 
2,12 28,12 20,38 0.9037 5,19 51,19 53,25 1.1906 
6,43 42,43 30,79 0.8806 6,6 36,6 46,36 1.1850 
16,35 48,35 38,67 0.8766 20,13 34,13 38,27 1.0477 
13,48 16,48 15,51 0.9835 18,45 30,45 34,57 1.0857 
10,8 49,8 36,47 0.9372 18,38 62,38 76,82 1.1325 

 

V. DISCUSSION 

Only linear dependencies of variables have been assumed 
for simplicity in the examples presented. If there is an 
occurrence of a non linear dependence of the input data, the 
square power exponent variables would extend combination 
polynomials of neurons and applied also as competent 
derivative terms (20). The denominators of (20)(21) result from 
the partial derivatives of the complete DE term polynomials of 
numerators. The root square (or power) functions are likely not 
involved into fractions (21), if the differences of values of input 
variables are not too big (in case a real data model) as occurred 
in presented examples. 

( )
( ) 21

21
2

2
1

21522110

4
1

2
2

2
18

2
2172

2
16215

2
24

2
1322110 ),(

xx

xxf

xxbxbxbb

xxaxxaxxaxxaxaxaxaxaa
y
i

∂∂
∂

=
+++

++++++++
=

(20) 

2
1

21
2

2
22210

215
2
24

2
1322110 ),(

x

xxf

xbxbb

xxaxaxaxaxaa
y
j

∂

∂
=

++

+++++
=

   (21) 

 
According to  (12)-(15) it is possible to define higher 

degree partial derivations of 2-variable compound function 
F(x,y)=f(u,v) (22) [6]. As the variables of the D-PNN’s inner 
functions u=φ(x1,y1) and ψ=(x2,y2) are different the 2

nd , 3rd and 
5th terms of eq. (23) are = 0 (as the partial derivation of ψ by x1 
is = 0). Likewise the terms of (24) and (25) do [6]. 

)],(),,([),(),( yxyxfvufyxF ψϕ==  u=φ(x, y) v=ψ(x, y)         (22) 

 

2

2

2

22

2

222

2

2

2

2

2
xv

f

xu

f

xv

f

xxvu

f

xu

f

x

F

∂

∂
⋅

∂

∂
+

∂

∂
⋅

∂

∂
+








∂
∂

∂

∂
+

∂
∂

⋅
∂
∂

∂∂

∂
+








∂
∂

∂

∂
=

∂

∂ ψϕψψϕϕ  (23)  

2

2

2

22

2

222

2

2

2

2

2
yv

f

yu

f

yv

f

yyvu

f

yu

f

y

F

∂

∂
⋅

∂

∂
+

∂

∂
⋅

∂

∂
+









∂
∂

∂

∂
+

∂
∂

⋅
∂
∂

∂∂

∂
+









∂
∂

∂

∂
=

∂

∂ ψϕψψϕϕ  (24) 

yxv

f

yxu

f

yxv

f

yxyxvu

f

yxu

f

yx

F

∂∂

∂
⋅

∂

∂
+

∂∂

∂
⋅

∂

∂
+

∂

∂
⋅

∂

∂
⋅

∂

∂
+










∂

∂

∂

∂
+

∂

∂

∂

∂

∂∂

∂
+

∂

∂
⋅

∂

∂
⋅

∂

∂
=

∂∂

∂

ψϕψψ

ψϕϕψϕϕ

22

2

2

2

2

22

   (25) 

A real data example might solve the weather forecast based 
on some trained data relations, which are used for calculating 
the next state of a system. Let’s take several types of variables 
(e.g. pressure, damp, temperature) partly describing states of 
this very complex system. The input vector of the D-PNN is 
formed by values of these variables in defined matrix 
coordinates of a meteorological map. The training data set 
includes definite states of a time interval and desired network 
outputs. The output could mark the weather forecast as 
“rainfalls”= 1, “cloudy” = 2, “sunshine” = 3. There can 
naturally arise possible transient states (e.g. 1.4). The output is 
computed for 1 locality of the map and could also predict the 
atmospheric pressure or another quantity. 

VI. CONCLUSION 

Artificial neural networks in general respond to related 
patterns with a similar output. They identify input patterns on 
the bases of their relationship. Likewise, the identification of 
unknown dependencies of the data variables could also be 
considered. This could be regarded as a pattern of abstraction, 
similar to that utilized by the human brain, which applies the 
approximation with time-delayed periodic activation functions 
of biological neurons in high dynamic system of behavior. D-
PNN is a new type of neural network, which performs 
identification based on any unknown generalized relations of 
input variables. D-PNN forms its functional output as a 
composition of differential equation terms (which describe a 
system of dependent variables) from rational integral functions. 
The problem of the multi-layered D-PNN construction reside 
creates every partial combination term for a complete DE in 
utilizing some fixed low combination degrees (2, 3), while the 
amount of variables is as a rule higher. 

REFERENCES 

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Bratislava, 2002 (in  Slovak). 

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[4] R. Rychnovský, J. Výborná, Partial differential equations and some of 
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[5] J. Kuneš, O. Vavroch, V. Franta, Principles of modelling, SNTL Praha, 
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[6] I. Kluvánek, L. Mišík, M. Švec, Matematics I., II., SNTL Bratislava, 
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[7] S. Das, A. Abraham, A. Konar, “Particle swarm ptimization and 
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[8] B. B. Misra, S. Dehuri, P.K. Dash, G. Panda, “A reduced and 
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[9] L. Zjavka, “Generalization of patterns by identification with polynomial 
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[10] L. Zjavka, “Construction and adjustment of differential polynomial 
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