BIBECHANA
Vol. 20, No. 3, December 2023, 267-274

ISSN 2091-0762 (Print), 2382-5340 (Online)
Journal homepage: http://nepjol.info/index.php/BIBECHANA

Publisher: Dept. of Phys., Mahendra Morang A. M. Campus (Tribhuvan University) Biratnagar

Machine learning driven prediction of lattice constants
in transition metal dichalcogenides

Bhupendra Sharma1, Laxman Chaudhary1,
Rajendra Adhikari2 Madhav Prasad Ghimire1,∗

1Central Department of Physics, Tribhuvan University, Kirtipur, 44613,
Kathmandu, Nepal.

2Department of Physics, Kathmandu University, Dhulikhel, Kavre, Nepal.
∗Corresponding author. Email: madhav.ghimire@cdp.tu.edu.np

Abstract
Machine learning represents an emerging branch of artificial intelligence, centering on the
enhancement of algorithms in computer programs through the utilization of data and the accu-
mulation of research-driven knowledge. The requirement for artificial intelligence in materials
science is essential due to the significant need for innovative high-performance materials on
a large scale. In this report, the gradient boosting regression tree model of machine learning
was applied to predict the lattice constants of cubic and trigonal MX2 systems (M=transition
metal and X=chalcogen atoms). The theoretical/experimental values of the materials were
compared to the predicted values to calculate the standard errors such as RMSE (root mean
square error) and MAE (mean absolute error). The features used to predict lattice constants
were ionic radius, lattice angles, bandgap, formation energy, total magnetic moment, density
and oxidation states. The features versus contribution barplot has been drawn to reveal the
contribution level of each parameter in the degree of [0,1] to obtain the predictions. This
report provides a precise account of the prediction methodology for lattice parameters of the
transition metal dichalcogenides family, a process that was previously not reported.

Keywords
Machine learning, Artificial Intelligence, Gradient Boosting Regression, Gradient Descent, RMSE,
MAE.

Article information
Manuscript received: August 18, 2023; Accepted: September 23, 2023
DOI https://doi.org/10.3126/bibechana.v20i3.57732
This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons.
org/licenses/by-nc/4.0/

1 Introduction

The crucial advancement of human history has
been manifested by the development of new mate-
rials. So a vast amount of data has been collected
throughout the centuries in the discipline of mate-
rial science. The database of the previous experi-
ments can be used to expedite innovations. The rise

of artificial intelligence (AI) ushers a new genesis
in the development of materials. AI works on the
basis of complex multilayer neural networks with
magnificent data mining ability. The fusion of ma-
terial science and AI methods are used to find the
complex relationship between different parameters,

267

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https://doi.org/10.3126/bibechana.v20i3.57732
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Bhupendra Sharma et al./ BIBECHANA 20 (2023) 267-274 268

predict the particular properties of materials and
improve the material characterization techniques.
The most favorable branch of AI in material sci-
ence is Machine learning (ML) [1].

Recently, ML is turning out into a robust ap-
proach to study the materials with extremely fast
and economical means. It works on the basis of
neural networks (NN) which performs the predic-
tion of required variables on the basis of training
data. The first principle calculation of materials
could now be enhanced by ML. We specifically de-
sign ML models to generate predictions based on
the correlations of the database through statistical
and probabilistic methods [2].

On the basis of nature of data labeling in mate-
rial science, ML can be divided into two categories:
Supervised learning and Unsupervised learning.
Supervised learning is a ML approach which is used
to evaluate an unknown mapping from known sam-
ples where the output is labeled (examples are clas-
sification and regression). This is like ‘y’ (depen-
dent variable) is predicted with the knowledge of
‘x’ (independent variable) trained previously [3].

In classification, there are p predictor variables
x1, x2...xp on which takes values 1, 2, ..., k are
trained. Our motive is to find a model to predict
the values of ’y’ from new values of ’x’. The classi-
fication tree algorithm is used to identify the cate-
gorical target variable of the most probable “class”
[4]. Also in regression models, the dependent vari-
able is estimated due to the range of independent
variable . The general linear regression model is:
y = β0 + β1x+ ε
where y is the dependent variable and x is inde-
pendent variable. β0 is the constant term which is
the intercept of the regression line on the vertical
axis. β1 is regression coefficient which is actually
the slope of the regression line. ε is the random
error which can be used to express the effect of ran-
dom factors on x [5].

Meanwhile, in unsupervised learning only input
samples are provided to the learning system to make
cluster and to estimate probability density function.
Hence, the main goal of this learning system is to
analyze the data, recognize a pattern and finally
structure within the available set of data.

Figure 1: Basic architecture of supervised learning.

Figure 2: Schematic picture of stepwise machine learning.

1.1 Process of Machine Learning

The major steps involved in the process of ML
are: [6]

• Identify the problem you want to solve using
ML

• Collection of training data for the process

• Selection of ML model

• Preparation of accumulated data to train in
ML model

• Test your ML system with test data



Bhupendra Sharma et al./ BIBECHANA 20 (2023) 267-274 269

• Validation and improvement of the ML
model. Usually, we need to search more train-
ing data during this iterative loop.

1.2 Density Functional Theory to Machine
Learning

Recently, we have entered the fourth paradigm of
science called data-driven science led by the ex-
periments and simulations. The influence of this
paradigm has led to the rise of new field called ma-
terial informatics. The objective of material in-
formatics is to discover the relationship between
known standard features and materials properties
such as structure, symmetry, composition, and
properties of the constituent elements. The im-
perceptible feature property relationships beyond
human capacities are recognized through ML pro-
cess [7].

A combination of density functional theory and
machine learning techniques provide a practical
method to excavate feature-property relationships
much more efficiently than by DFT or experi-
ments [8]. As the fundamental mathematical model
of DFT works only for ground state density so
the study of excited states is limited within this
method. The strongly correlated systems, such as
d-electron in transition metal oxides (TMO) are
solved with supplemental theories like Hubbard pa-
rameter (U). To extend the proficiency of DFT,
various auxiliary code based models are integrated

with DFT codes such as VASP, Quantum Espresso,
and WIEN2k. These methods are vastly compu-
tationally demanding but consistent and reliable.
Hence, the abundant database obtained from suc-
cessful DFT platforms can be fitted into appropri-
ate ML models to steer the discovery of complex
properties of materials [7].

1.3 Transition Metal Dichalcogenides

Two dimensional transition metal dichalcogenides
(2D-TMDs) are layered materials with robust in-
plane bonding and weak van der Waals interactions
between the planes which enables the exfoliation
into thickness of two dimensional layers with single
unit cell [9]. TMDs are an emerging class of materi-
als with highly attractive properties such as atomic-
scale of thickness, direct bandgap and strong spin
orbit coupling. Hence these materials accelerate
the fundamental studies of novel physical phenom-
ena with applications ranging from nano-electronics
and nanophotonics to sensing and stimulation at
the nanoscale [10]. The different properties of bulk
TMDs ranges from insulators such as HfS2, semi-
conductors such as MoS2 and Ws2, semimetals such
as WTe2 and TiSe2, to true metals such as NbS2
and VSe2. The properties are mostly preserved
with the exfoliation of such materials into mono
or few layers with supplemental characteristics that
emerges from confinement [11].

Figure 3: Bulk crystal structure of MoS2.



Bhupendra Sharma et al./ BIBECHANA 20 (2023) 267-274 270

2 Materials and Methodology

In this section, the methodology and the required
programming model used in our work has been de-
scribed.

2.1 Data Mining

It is necessary to collect qualitative data for re-
searchers to create the model for specific problems.
Open source databases such as Materials Project
[12] and Open Quantum Materials Database [13]
are one of reliable and representative databases.
These databases could be used to extract the data
set of the materials of our concern.

Once the collection of data is done, we need to
clean the problems in data such as the data contain-
ing abnormal or redundant values. This is called
data cleaning. The data cleaning steps are:

• Data Sampling

• Processing of abnormal value

• Discretization of data

• Data Normalization

Data sampling allows us to create prediction model
of high performance with less data. Thereafter, the
removal of abnormal values maintains the accuracy
of the model. The continuous features are reduced
by data discretization. Finally, data normalization
is used to organize the columns and tables on the
basis of dependencies seen in data. After these
steps, the data set is split into training and test-
ing sets to subject it into the ML model [14].

2.2 Gradient Descent

The objective of Gradient Descent Algorithm is to
minimize a differentiable cost function in certain
number of iterations [15]. If y = mx + c is regres-
sion model then m and c are parameters. So, to
find the optimum value of y, m (slope) and c (y-
intercept) must be minimized. Then the cost func-
tion becomes:

J(m, c) =

n∑
i−1

(yi − (mx+ c))2 (1)

This cost function is used to optimize the parame-
ters m and c with following relation.

mnew = mold − ρ
∂J(m, c)

∂m

∣∣∣∣∣
(m′,c′)

(2)

where, ρ is the learning rate and (m’, c’) is initial
guess. Similarly,

cnew = ccold − ρ
∂J(m, c)

∂m

∣∣∣∣∣
(m′,c′)

(3)

So the cost function has to be minimized by mov-
ing in opposite direction of the gradient. The it-
eration process will be repeated until we reach the
minimum cost function. Finally, we obtain the op-
timized parameters for best fit in our regression
model [16].

2.3 Cross-Validation

Cross-validation technique is a resampling method
in ML where the observation set is split into two
subsets. The first subset is called training set and
the second one is called test set. The training set is
exercised to find a proper function whose predictive
capacity is based on the prediction error. Predic-
tion error is calculated by applying the result of
training set into the test set.

In this report, we have used k-fold cross valida-
tion method because it estimates the model in finer
way by training and evaluating it k-times enhanc-
ing the model performance. In this method, the
dataset is partitioned into k subsets (folds), then
the function is trained on the k-1 subsets. Finally,
the prediction error is estimated on each k-1 sub-
sets with the help of the remaining set [17]. The
other validation methods such as Holdout valida-
tion that splits whole data into training and vali-
dation set could be sensitive to the initial random
split and the evaluation can biased depending on
the split. Also, we opt k-fold cross-validation as
our preferred method for model evaluation as it is
most suitable for smaller datasets. Ultimately, the
choice of validation technique is chosen on the basis
of the characteristics of our data and the research
objectives.

2.4 Gradient Boosting Regression Model

We have used Gradient Boosting regression tree
(GBRT) to predict the lattice parameters of cubic
and trigonal TMDs. A GBRT model minimizes the
prediction errors through boosting technique which
compiles the set of weak models to construct a sin-
gle strong model. In this technique, the prediction
error is minimized sequentially by generating new
decision trees. Decision tree compares the values of
root attribute with the real data-set attribute and
on the basis of the comparison, it follows the branch
and moves to the next node. The sequential process
is basically a functional gradient descent where the
prediction is optimized with the addition of new
tree at every step, in order to minimize the cost
function [18].

In this work, we have predicted the lattice con-
stants (a, b and c) using other properties of MX2

system such as ionic radius, lattice angles, band
gap, formation energy, total magnetic moment,
density and oxidation number using GBRT model.
The hyperparameters of GBRT model used here are



Bhupendra Sharma et al./ BIBECHANA 20 (2023) 267-274 271

max-depth and n-estimators. The max-depth sets
the maximum depth of nodes in a tree which is set
to 4. The n-estimator chooses the number of trees
to be considered in our model which is set to be
500. The 3-fold cross validation is applied to the
model with shuffle on mode. This means that the
dataset has been split into three subsets. Two of
them were used for training where remaining one
was used for testing the model and the data are
trained and tested in shuffle mode.

3 Results and Discussion

The interpretation of the accuracy of our model is
measured on the basis of Root mean square error
(RMSE) and mean absolute error (MAE). For n
observations y with n corresponding model predic-
tions ŷ (yi, i=1,2...n), RMSE and MAE modeled by
Hodson [19] are as below:

RMSE =

√√√√ 1

n

n∑
i=1

(yi − ŷi)2 (4)

MAE =
1

n

n∑
i=1

|yi − ŷi| (5)

Figure 4: Barplot (i) shows the features used and their roles in the model to predict a of the cubic MX2

system (ii) shows the features used and their roles in the model to predict a of trigonal MX2 system
(iii) shows the features used and their roles in the model to predict b of trigonal MX2 system (iv) shows
the features used and their roles in the model to predict c of trigonal MX2 systems. The figures were
generated in Jupyter Notebook using Python codes.

3.1 Lattice Constants Prediction for Cubic
Systems

Twenty one cubic TMDs that were available in ma-
terials project website has been used. We used

GBRT algorithm to predict the lattice constant of
such system. Since in cubic system a = b = c and
α = β = γ, so it is sufficient to predict one lattice
constant for this structure.

Average RMSE and MAE from three validations
are 2.62Å and 0.824Å, respectively. Figure 4 (i)
shows that the order of importance of the parame-
ters is density, total magnetic moment, ionic radius
of first element, formation energy, oxidation state
of second element, ionic radius of second element,
oxidation state of first element, and bandgap (de-

noted as density, totalMag, ra(pm), formE, OXb,
rb, OXa, and band gap, respectively, in Figure 4.
So it is seen that the density has the most impor-
tant role in pattern recognition by the model, and
band gap has the least. This is because almost all
data of cubic systems possess zero band gap.



Bhupendra Sharma et al./ BIBECHANA 20 (2023) 267-274 272

Table 1: Comparison table of predicted lattice constants versus their experimental/theoretical values in
cubic MX2 system.

Material Experimental/Theoretical (a) Predicted (a’)

MnTe2 6.69 6.17
CoS2 5.51 5.58
CoSe2 5.84 5.89
NiS2 5.61 5.71
NiSe2 5.94 6.13
YS2 7.83 5.87
CuSe2 6.16 6.31

3.2 Lattice Constants Prediction for Trig-
onal Systems

Thirty-one trigonal TMDCs have been taken from
project materials to predict the lattice constants of
such systems using a GBRT model. In the trigonal
crystal system, the relation between major lattice
parameters (lattice constants and lattice angles) is
given by a= b ̸=c, α = β = 900 and γ = 1200.
The calculated predictions of lattice constants are
shown in Table 2.

Figure 4(ii) shows that the prediction of a is
mostly influenced by the ionic radius to draw the
pattern relation, whereas the lattice angles have the

least important role, primarily because the angles
of all the systems are identical. The average stan-
dard errors after three validations are: RMSE =
0.0542 Å and MAE = 0.205 Å.

In Figure 4(ii) and (iii), a and b are equal for
the trigonal system, so almost identical errors are
seen in predicting b with the same feature roles.

In the prediction of c, formation energy has
shown the most important role in pattern recogni-
tion, while density contribution is in second place,
as shown in Figure 4(iv). The average standard er-
rors after three validations are found to be RMSE
= 0.547 Å and MAE = 0.462 Å.

Table 2: Comparison table of predicted lattice constants versus their experimental/theoretical values in
trigonal MX2 systems.

Material a a′ b b′ c c′

FeTe2 3.75 3.75 3.75 3.66 5.84 5.23
VTe2 3.66 3.64 3.66 3.55 6.95 6.13
CoTe2 3.79 4.02 3.79 4.00 5.56 5.19
CuTe2 4.02 3.90 4.02 3.90 5.10 5.24
ZnTe2 4.19 3.83 4.19 3.83 5.20 6.12
ZrTe2 3.98 3.79 3.98 3.79 7.00 6.73

4 Conclusions

In this work, the prediction of lattice constants
of cubic and trigonal MX2 systems has been done
through a Gradient Boosting Regression model of
machine learning. We have predicted a, b, and c on
the basis of patterns or relationships drawn through
training the algorithm with features such as ionic
radius of atoms, lattice angles, bandgap, formation
energy, total magnetic moment, density, and oxida-
tion states. In the cubic system, the RMSE and
MAE in the prediction of lattice constants were
found to be 2.62Å and 0.824Å, respectively. In
the trigonal system, the RMSE of a = b and c were
0.0542Å and 0.547Å, respectively. Whereas, the
MAE of a = b and c were 0.205Å and 0.462Å, re-

spectively.

The barplot of cubical systems shows that the
contribution of density is the most significant in
prediction, followed by total magnetic moment and
ionic radius. Whereas, the barplot of the trigonal
system indicates ionic radius as the major contribu-
tor in the prediction of a = b, and formation energy
as the major contributing parameter to predict c.

DFT calculations and experiments are discov-
ering a plethora of materials day by day. Hence,
a large collection of data can be acquired in the
field of 2D-TMDs in the near future. The conse-
quence is that the huge database could be used in
machine learning to accelerate materials discovery
with higher precision in the least amount of time.
Thereafter, not only lattice constants but every lat-



Bhupendra Sharma et al./ BIBECHANA 20 (2023) 267-274 273

tice parameter we desire can be predicted with ex-
treme accuracy.

Applications

The cubic and trigonal TMDs are one of most
stable, extensively researched and then advanta-
geous form of 2D materials. Predicting their lat-
tice parameters using machine learning has numer-
ous practical implications across materials science
and related fields. These predictions can expedite
materials discovery by guiding researchers toward
promising TMD combinations, impacting areas like
catalysis and energy storage. Moreover, the precise
control offered by these predictions aids in designing
TMDs for electronic and optical applications, in-
fluencing the development of semiconductors, pho-
todetectors, and optoelectronic devices. This can
lead to advancements in electronics, photonics, and
telecommunications. They are also invaluable for
materials characterization, aiding experimentalists
in structural analysis techniques. Experimentalists
can use these predictions as reference values during
structural analysis techniques such as X-ray diffrac-
tion and electron microscopy to validate the quality
of synthesized samples. Overall, machine learning’s
role in TMD lattice parameter prediction acceler-
ates innovation, improves materials properties, and
reduces costs across a broad spectrum of applica-
tions.

Ethical considerations

In our study, we adhere to ethical guidelines, even
when utilizing data sourced from the Materials
Project. Although the data is not confidential, we
uphold ethical standards because our research can
have societal implications. We adhere to the Ma-
terials Project’s data privacy policies, and we take
deliberate steps to prevent bias through the careful
selection, analysis, and transparency of data. For
additional information about the Materials Project,
please visit the website [12].

Acknowledgments

This work was supported by a grant from
UNESCO-TWAS and the Swedish International
Development Cooperation Agency (SIDA) with
TWAS Research Grant Award No. 21-377
RG/PHYS/AS-G. The views expressed herein do
not necessarily represent those of UNESCO-TWAS,
SIDA or its Board of Governors. Part of this work
was also supported by the University Grants Com-
mission, Nepal under UGC Grant No. CRG-78/79
S&T-03. We acknowledge NVIDIA for providing
the license package of Deep learning, a part of ma-
chine learning.

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https://oqmd.org/

	Introduction
	Process of Machine Learning
	Density Functional Theory to Machine Learning
	Transition Metal Dichalcogenides

	Materials and Methodology
	Data Mining
	Gradient Descent
	Cross-Validation
	Gradient Boosting Regression Model

	Results and Discussion
	Lattice Constants Prediction for Cubic Systems 
	Lattice Constants Prediction for Trigonal Systems

	Conclusions