# Machine Learning for High-entropy Alloys: Progress, Challenges and
Opportunities 111This manuscript has been co-authored by UT-Battelle, LLC,
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Xianglin Liu

Peng Cheng Laboratory
Oak Ridge National Laboratory
Jiaxin Zhang

Oak Ridge National Laboratory
Zongrui Pei

New York University
Oak Ridge National Laboratory
<EMAIL_ADDRESS>

###### Abstract

High-entropy alloys (HEAs) have attracted extensive interest due to their
exceptional mechanical properties and the vast compositional space for new
HEAs. However, understanding their novel physical mechanisms and then using
these mechanisms to design new HEAs are confronted with their high-dimensional
chemical complexity, which presents unique challenges to (i) the theoretical
modeling that needs accurate atomic interactions for atomistic simulations and
(ii) constructing reliable macro-scale models for high-throughput screening of
vast amounts of candidate alloys. Machine learning (ML) sheds light on these
problems with its capability to represent extremely complex relations. This
review highlights the success and promising future of utilizing ML to overcome
these challenges. We first introduce the basics of ML algorithms and
application scenarios. We then summarize the state-of-the-art ML models
describing atomic interactions and atomistic simulations of thermodynamic and
mechanical properties. Special attention is paid to phase predictions, planar-
defect calculations, and plastic deformation simulations. Next, we review ML
models for macro-scale properties, such as lattice structures, phase
formations, and mechanical properties. Examples of machine-learned phase-
formation rules and order parameters are used to illustrate the workflow.
Finally, we discuss the remaining challenges and present an outlook of
research directions, including uncertainty quantification and ML-guided
inverse materials design.

Keywords: High-entropy alloys; Machine learning; Atomistic simulations;
Physical properties; Alloy design

###### Contents

  1. 1 Introduction
    1. 1.1 High-entropy alloys
    2. 1.2 Challenges in modeling high-entropy alloys
    3. 1.3 Merits of machine learning for high-entropy alloys
    4. 1.4 Uniqueness and structure of this review
  2. 2 How to utilize machine learning for high-entropy alloys?
    1. 2.1 Application identification
    2. 2.2 Representative machine-learning algorithms
    3. 2.3 Data collection
      1. 2.3.1 Data for atomistic simulations
      2. 2.3.2 Data for physical-property models
      3. 2.3.3 Data augmentation
    4. 2.4 Model selection
    5. 2.5 Model training
    6. 2.6 Result analysis
  3. 3 Atomic interaction models
    1. 3.1 Symmetry-preserving descriptors
    2. 3.2 Machine-learning potentials
      1. 3.2.1 High-dimensional neural network potential (HDNNP)
      2. 3.2.2 Gaussian approximation potential (GAP)
      3. 3.2.3 Spectral neighbor analysis potential (SNAP)
      4. 3.2.4 Moment tensor potential (MTP)
    3. 3.3 Effective rigid-lattice Hamiltonian
  4. 4 Machine-learning accelerated atomistic simulations
    1. 4.1 Monte-Carlo simulations
    2. 4.2 Molecular-dynamics simulations
    3. 4.3 Hybrid MC/MD simulations
  5. 5 Physical-property prediction
    1. 5.1 Prediction of solid solutions and multi-phases
    2. 5.2 Machine-learning informed new rules for phase predictions
    3. 5.3 Machine-learning based order parameters
    4. 5.4 Mechanical-property prediction
  6. 6 Challenges and Outlook
    1. 6.1 Challenges
    2. 6.2 Uncertainty quantification
    3. 6.3 End-to-end (automatically learned) descriptors
    4. 6.4 Machine-learning guided inverse materials design
  7. 7 Conclusions
  8. 8 Acknowledgements

##  1 Introduction

###  1.1 High-entropy alloys

The conventional strategy of alloying is to add a small amount of property-
enhancing elements into one primary metal. Humans have used this technique for
thousands of years, exemplified by some of the most widely used materials in
human history, such as bronzes, steels, and aluminum alloys. However, the
discovery of the single-phase multicomponent high-entropy alloys (HEAs) in
2004 [1, 2] introduced a different alloying strategy. While conventional
alloys are comprised of one base element with a few additional elements of
much lower concentrations, in HEAs, multiple-principal elements are mixed to
obtain a single-phase random solid solution. These new classes of materials
are named HEAs [1, 3] due to their supposedly enhanced configurational
entropy, which was believed to be the cause of their stabilization. However,
the role of configurational entropy is not always as important as originally
assumed [3, 4, 5], therefore names such as multicomponent alloys, multi-
principal-element alloys, compositionally-complex alloys, or complex
concentrated alloys have also been widely used [6]. This new strategy of
mixing multiple principal elements opens the door to exploring the vast
uncharted space beyond the corners of chemical compositional diagrams. HEAs
have become one of the most exciting research directions in materials science,
and HEAs of unprecedented performance have been constantly emerging, as
demonstrated in Fig. 1 b. Since there are a number of excellent review papers
focused on HEAs [7, 8, 9, 10, 11, 12, 13], we have no intention to restate the
details therein, which is also not the main focus of this work.

Several key factors contribute to the enormous interest in HEAs. The first one
is their exceptional mechanical properties, such as the exceptional high-
temperature properties [14], enhanced strength, ductility and toughness at
cryogenic temperatures [15, 16], and most importantly the excellent
combination of strength and ductility [2, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26]. Some HEAs demonstrate the unusual capability of overcoming the strength-
ductility tradeoff [17, 20, 21]. Another reason is that HEAs also provide an
attractive platform to study deformation mechanisms related to chemical order
and disorder. The different occupations of lattice sites by the multiple
principal elements give rise to various degrees of randomness and order,
characterized by short-range order (SRO) [27, 28, 29] and long-range order
(LRO) parameters [30, 31], which have a significant impact on the physical
quantities crucial for plastic deformation, such as dislocation core
structures and generalized stacking fault energies [32, 33] [34]. Tuning the
degree of order and disorder can improve the mechanical properties of HEAs
[35, 36, 27, 28, 29] [37]. A third reason is their diverse and heterogeneous
microstructures, such as the polymorphology of stacking faults [38], twin
architectures [22], decoupling of between Shockley partials and stacking
faults [39], nanoscale phases [17], ultrafine-grained lamella structures [19],
and nano-precipitates [20, 25, 21, 40]. While these microstructures can
present in conventional alloys, the large number of elements in HEAs bring
more degrees of freedom to tune them. The high tunability is beneficial to
enhancing the mechanical properties of single-phase random HEAs. Last but not
least, the barely explored non-equiatomic concentrations in the vast, high-
dimensional compositional space of HEAs, demonstrate the huge potential for
materials design.

###  1.2 Challenges in modeling high-entropy alloys

While the increased principal elements bring huge opportunities to alloy
design, they also introduce significant challenges to the theoretical modeling
and simulations. The first one is the difficulty in constructing empirical
atomic interaction models, mainly due to the large number of chemical
interactions involved. For instance, for an $n$-component system, the number
of $m$-site interactions proportional to

$C(n,m)=\frac{n!}{m!(n-m)!}=\frac{n(n-1)\cdots(n-m+1)}{m(m-1)\cdots 1}.$

For a five-element system, only considering the pair interactions within the
first four coordination shells already give $4\times C(5,2)=40$ different
interactions. Considering the higher-order interactions would substantially
further increase the number of interactions, making the traditional cluster
expansion method [41, 42, 43, 31] and embedded atom method (EAM) [44] prone to
overfitting. The second challenge occurs in the first-principles simulations.
It is well-known that density functional theory (DFT) simulations are
generally computationally expensive. However, they are more prohibitive for
HEAs due to the necessity of using large supercells (up to more than 1000
atoms) to represent the non-stoichiometric compositions, complex order-
disorder behaviors, microstructures, and extended dislocation core structures
[45] [34]. The computational cost is even more demanding for first-principles
thermodynamics simulations, where many configurations must be evaluated, each
calculated from the expensive DFT. For instance, for a 250-atom CuZn
supercell, the first-principles Monte-Carlo simulation requires the self-
consistent (SCF) calculations of 600,000 DFT energies [46], which can easily
consume a million CPU core-hours. Finally, the traditional trial-and-error
method of materials design is too laborious for HEAs, therefore substitute
models are highly desirable to guide the efficient exploration of the vast
compositional space, such as phase formation rules [47] [48], strengthening
model [44, 49] and ductility criteria [50][51, 52]. Nevertheless, these
empirical rules are challenging to formulate for HEAs due to the large number
of chemical species involved.

###  1.3 Merits of machine learning for high-entropy alloys

The past decade has witnessed the explosive rise of machine learning (ML) and
its sub-domain deep learning [53]. This revolution has not only fundamentally
changed several domains of computer science, such as computer vision and
natural language processing, but also a plethora of scientific fields ranging
from particle signal detection [54], protein structure prediction [55],
medical image analysis [56], to the survey of the universe [57]. The success
of ML lies in its capability to describe complex patterns, and these complex
ML models can be systematically optimized by learning from the data. This
capability is further powered by readily available computing resources,
efficient algorithms, and big data collected from experiment or computation.
These features make ML a promising tool to address the aforementioned
challenges confronted by the theoretical modeling of HEAs. The exponential
growth of relevant publications reflects this trend (see Fig. 1 a). In
principle, given a sufficiently large high-quality data, the complex atomic
interactions can be well-captured by ML through the standard procedures of
training, validation, and testing.

Moreover, the calculation speed of ML models is typically much faster than DFT
and close to empirical potentials [58, 59, 60, 61]. This high efficiency
allows researchers to simulate materials with more than millions of atoms near
DFT accuracy [62, 63], way beyond the limit of conventional DFT. Such
capability is particularly important for understanding the extraordinary
mechanical properties in HEAs, originating from the complex ordering, defects,
and microstructures [64, 65] that demand accurate nanoscale simulations.
Besides simulations, ML also provides a powerful tool to build prediction
models for material properties. By learning from the data, ML can
automatically identify features or latent variables essential for achieving
the desired physical properties. These complex, nonlinear structure-property
relationships are challenging to formulate by human observation. Identifying
them allows researchers to explore the enormous design space of HEAs more
efficiently. The above merits of ML has ushered in a new data-driven paradigm
to the research of HEAs [66, 67], as demonstrated in its success in the
atomistic simulations [68, 69, 70], physical property prediction [48][71], and
materials design [72, 73, 74].

Figure 1: The progresses in high-entropy alloys and machine learning. (a)
Annual publications retrieved with the keywords “machine learning material”
and “machine learning high-entropy alloys” (inset plot) from ScienceDirect
(blue) within the subject area of materials science, Nature Portfolio journals
(green), and Physical Review journals (red). Search date: July 26, 2022. (b)
Time table of a few representative high-performance HEAs.

###  1.4 Uniqueness and structure of this review

This review focuses on the application of ML for high-entropy alloys (HEAs).
The progress, challenges, and opportunities of this specific topic are within
the scope of the review. While several excellent reviews have already been
available on the broader topic of ML for materials science [75, 76, 77, 78,
79, 80, 67, 66], there is a lack of comprehensive reviews for the exciting
progress and huge potential of ML for HEAs. For instance, Drodola reviewed the
applications of ML for alloys in a recent publication [66], which focused on
using artificial neural networks for alloy design, processing, and
characterization. Another review on ML for alloys was presented by Hart et al.
[67], where they summarized the current state of machine-learning-driven alloy
research. Both review papers summarize ML applications for alloys in general.
Still, ML’s unique challenges and opportunities for HEAs are not thoroughly
discussed. Given the extensive attention that HEAs have attracted, such a
timely, focused review is needed. To be specific,

  * •

We present a comprehensive review on the interdisciplinary topic of ML for
HEAs, discuss the basic methods and models heuristically, and summarize the
important applications to illustrate the advantages of ML for HEAs;

  * •

We identify challenges in the theoretical modeling of HEAs, and illustrate how
ML can help tackle these problems from both microscopic and macroscopic
perspectives;

  * •

We compare the strengths and shortcomings of different ML methods for HEAs,
summarize the remaining challenges of ML methods, and discuss the
opportunities accompanied by these challenges in the future.

This review is intended to pinpoint the opportunities in ML for HEAs and
provide guidance to researchers interested in exploring this new field. To
this end, we will introduce the ML methods, survey their applications in
multicomponent alloys, and elucidate the remaining challenges. This review is
organized as follows: First, an introduction to ML for HEAs is given,
including application identification, data generation, common algorithms,
model training, and results analysis. Next, popular ML potentials for
atomistic simulations are discussed. The following three sections focus on the
recent progress in applying ML to HEAs, including atomistic simulations,
property predictions, and materials design. Finally, the challenges and
opportunities of ML for HEAs will be discussed in the outlook section.

##  2 How to utilize machine learning for high-entropy alloys?

The ML model’s general workflow is illustrated in Fig. 2. While ML is
generally a versatile and powerful tool, one should keep in mind that it only
shines for a certain subset of problems. Specifically, ML is most effective
for problems with abundant high-dimensional data and well-defined objectives.
Therefore, the first step for utilizing ML for HEAs is application
identification [81], i.e., confirming that the problem can be converted to an
ML one. After that, appropriate ML algorithms should be chosen to solve the
specific ML problem. The data are collected and generally split into three
subsets, i.e., (i) the training set used for model training, (ii) the
validation set for model selection, and (iii) the test set for model
performance evaluation. Finally, a well-trained ML model should capture the
underlying physics. Good-performance models render it possible to extract the
knowledge by inspecting their parameters directly or through dimension
reduction techniques [82].

Figure 2: The critical steps in a typical ML application for HEAs. (a)
Application identification. Typical applications include accelerating
simulations, predicting physical properties, and extracting hidden physics.
(b) Selection of appropriate machine learning algorithms, such as neural
networks (NN), support vector machine (SVM), Gaussian process (GP),
classification and regression tree (CART), and generative adversarial network
(GAN). (c) Data collection. The data can be obtained from a database and
augmented with techniques such as GAN. (d) Model training. A typical example
is stochastic gradient descent to optimize the parameters in neural networks.
(e) Model selection. Grid search and Bayesian optimization can be used. (f)
Result analysis. The model parameters can be analyzed with dimension-reduction
techniques.

###  2.1 Application identification

One of the most common ML applications for HEAs is to build surrogate models.
Surrogate models are used to approximately model a set of input-output data
when the actual relationship is unknown or difficult to deduce. Currently, the
applications of these surrogate models can be roughly grouped into (i)
atomistic simulations on the atomic scale and (ii) physical property
predictions on the macro scale. Here we refer to the corresponding ML models
as atomic interaction models (AIMs) and bulk property models (BPMs). The
applications are compared in Tab. 1. Specifically, AIM and BPM tackle
materials science problems from different time and length scales. AIM follows
the spirit of first-principles methods. However, instead of subatomic
electrons and nuclei, as in DFT, the input feature of AIM is the arrangement
of different atoms. In the language of the renormalization group, the electron
degree of freedom has been “integrated out”. Compared to BPM, one major
benefit of AIM is its high transferability. Well-trained AIM can be employed
to evaluate many different physical quantities, such as the formation energy,
elastic constants, stacking fault energies, and phonon spectra [59, 60]. By
comparison, the advantage of BPM is that it directly predicts the target
physical properties without the involvement of expensive simulations. Another
key difference between AIM and BPM is the fidelity requirement: AIMs generally
have a higher requirement on the accuracy of the model prediction than BPMs.
For instance, a nonphysical force prediction in one step in molecular dynamics
may lead all the following simulations to a completely wrong trajectory.
Therefore, it is important to tailor physical information into AIMs, such as
the materials’ underlying translational, rotational, and permutational
symmetry.

Table 1: Comparison of ML surrogate models for atomistic simulations and bulk
property predictions.

|  Atomistic Simulation |  Bulk Property Prediction
---|---|---
Role of ML |  Providing empirical potentials to accelerate simulations |  Predicting measurable or calculable physical properties
Targets |  Atomic energies, forces, and stresses |  Phase formation, crystal structure, elastic constants
Data source |  First-principles calculations |  First-principles calculations, experimental data
Data generation |  Computationally expensive (hundreds to millions of CPU hours) |  Experimentally expensive ($>$ hundreds of experiments)
Descriptors |  Invariant representation of atomic environment |  Properties of chemical elements, relevant physical properties
Models/Algorithms |  Physical Descriptors + ML |  Traditional ML algorithms (e.g., support vector machine, decision trees, Gaussian process) and neural networks
Model inference |  Needed for each atom at every simulation step |  Needed for exploring new materials

Acronyms: ML- machine learning; CPU-central processing unit.

In addition to building surrogate models, ML can also be employed to draw
physical insights from extensive data. To this end, reducing data dimensions
and extracting important features are critical since the physical rules are
more evident in small dimensions. For some ML models, such as decision trees
and shallow neural networks [83, 84], essential features can be identified by
inspecting the weight parameters of the models. Unsupervised learning is also
a widely used technique to extract hidden physics. Unsupervised learning does
not require labor-intensive data labeling, which is one of its unique
advantages. Instead, it explores the patterns and structures in the unlabeled
dataset. A typical example of unsupervised learning is clustering, which
divides the data into groups according to their similarity in the feature
space. Clustering can be used as a dimension reduction technique to classify a
large number of features into a few groups. With the grouped features, the
relation between the desired properties and a few labeled data can be
established straightforwardly [85]. Another excellent example is the
variational autoencoder (VAE) [86]. It can significantly reduce the data size
and simultaneously keep the essential information in the latent space with
only very few dimensions, usually two, for better visualization. Assisted by
this advantage of VAE, Yin, Pei, and Gao proposed a VAE-based order parameter
to describe the phase transitions in HEAs [70]. The new order parameter and
VAE model are based on many chemical configurations generated using Monte-
Carlo simulations, informed by the interaction parameters also obtained from
ML models. The order parameters for phase transitions are one of the central
topics in condensed matter physics and materials science, so this also
demonstrates the incredible power of ML contributions to physics.

###  2.2 Representative machine-learning algorithms

Different ML algorithms have different strengths and shortcomings. Therefore,
it is crucial to choose the appropriate one according to the task objective
and dataset size. For example, when the dataset size is small, a simple linear
regression should perform better than a deep neural network model due to the
well-known bias-variance tradeoff [87]. There are already many excellent
introductions to ML algorithms [87, 53], so here we only briefly describe a
few representative ones.

Neural network (NN): Neural networks, also referred to as artificial neural
networks (ANNs), are the backbone of deep learning algorithms [88, 89]. A
neural network is comprised of multiple function layers, with each layer
composed of multiple neurons. A neuron receives inputs from its previous
layer, multiplies them (dot-product) with its weight parameters, adds the bias
term, and passes the result to the activation function to produce the output.
One simple example of a neural network is the multilayer perceptron, which is
a feed-forward network comprised of a few fully connected layers. Some other
frequently used neural network structures include the convolutional neural
network (CNN) [90], recurrent neural network (RNN) [91], and transformers
[92]. NNs are typically optimized with stochastic gradient descent (SGD)
algorithms, in which the backpropagation method is commonly employed to update
the model parameters, aiming to minimize the loss function for given data
samples. The loss function, by its name, measures the deviation from the
desired distribution. For supervised learning, the loss function typically
measures the difference between the model prediction and the true target
values. Choosing an appropriate loss function is very important for ML. For
regression tasks, the quadratic loss function is a common choice, while the
cross entropy loss is widely used for classification tasks.

Support vector machine (SVM): SVM is a widely used algorithm for separating
different groups of data [93]. The principle of SVM is to find a hyperplane
that maximizes the separation between the two groups. The hyperplane decides
which group a data point belongs to. In linear SVM, the decision plane
$\mathbf{w}^{T}\mathbf{x}+b=0$ that maximizes the separation is given by
minimizing $||\mathbf{w}||$ under the constraint
$y_{i}(\mathbf{w}^{T}\mathbf{x}_{i}+b)\geq 1$, for all data
$(\mathbf{x}_{i},y_{i})$, where $y_{i}\in\\{-1,1\\}$. As a result, the
decision plane only depends on the closest data points unaffected by the
others. Other than the above “hard margin” version, SVM can also be easily
extended to cases where the data are not linearly separable. A regularization
term is usually used to penalize the margin violation, which is called soft-
margin SVM. Moreover, the separation boundary in SVM can also be extended to
nonlinear cases by employing the so-called kernel method, which amounts to
implicitly projecting the data to a high-dimensional feature space. One common
choice for the kernel is the radial basis function (RBF), which takes the form
of a Gaussian function and has infinite feature space. Finally, multi-class
SVM can be formulated by decomposing the problem into multiple binary
classification problems other than binary classification.

Gaussian process (GP): GP is based on Bayes’ theorem, i.e., the posterior is
proportional to the product of prior and likelihood [94]. In GP regression,
each data $x$ is assumed to follow one Gaussian distribution, and nearby data
points $x$ and $x^{\prime}$ are related smoothly by the covariance function
(kernel). Due to the multivariate Gaussian distribution assumption, the
posterior can be easily expressed in terms of the covariance matrix of the
data points. Therefore, the training process essentially evaluates the
covariance matrix and its inverse, from which the probability distribution of
model parameters can be obtained. During the inference stage, the prediction
is obtained by averaging the output of all possible models. The most
attractive feature of GP is that it can estimate the probability distribution
of the model parameters, which makes uncertainty quantification
straightforward [95]. On the other hand, the Gaussian process is generally
computationally expensive for a large dataset ($>$ 10,000 data points), in
which case some sparsity technique is needed to reduce the cost by choosing a
sub-sample of the entire dataset [96]. Other than regression, GP can also be
used for classification tasks by appending a logistic or softmax function.
However, introducing a non-Gaussian function makes the model difficult to
solve, and methods such as the Laplace approximation are needed to avoid the
computationally intensive Monte-Carlo sampling.

Classification and regression tree (CART) A decision tree is an intuitive and
easy-to-interpret method [97]. It starts from a root node, and criteria are
chosen for the features to split the data into subsets, as represented by the
descendant nodes. This process continues until the termination condition is
reached. The splitting criteria are based on a randomness measure for the
subset. One widely used measure is the Gini impurity, defined as
$G=1-\sum_{i}p_{i}^{2}$, where $p_{i}$ is the ratio of class $i$ in the node.
In terms of the Gini impurity, the cost function for splitting the data into
$n$ subsets, i.e. Gini index, is given by $L=\sum_{i}\frac{m_{i}}{m}G_{i}$
where $m_{i}$ is the number of data in the $i$-th subset, and
$m=\sum_{i}m_{i}$. A single DT is a relatively weak predictor, and the
performance can be improved by combining a group of predictors, each trained
with a data subset from bootstrap aggregating (bagging). When each decision
tree uses a random set of features for splitting, such methods are called
random forest (RF), a widely used ensemble method [98]. A stronger predictor
can also be obtained by sequentially improving the performance; such an
ensemble method is known as boosting [99]. Typical examples include AdaBoost,
and Gradient Boosting [87].

###  2.3 Data collection

####  2.3.1 Data for atomistic simulations

The training datasets are generally obtained via DFT calculations for
constructing atomic-interaction surrogate models. A high-quality training
dataset for ML models should be homogeneous, sufficiently large, and
representative. By homogeneous, we mean the DFT energies are calculated using
the same method, with the same set of parameters, so that the energy
difference between data can truly reflect the impact of atomic configuration.
Important DFT parameters include the type of exchange-correlation functional,
the energy cutoff of plane waves, the mesh grid of k-points, etc. A
sufficiently large dataset is necessary for the model to capture the
underlying physics accurately. Finally, a representative dataset means the
sampled data points are evenly distributed in the configuration space. This
representativity is crucial for an accurate surrogate model since ML
algorithms are usually much more reliable doing interpolation than
extrapolation.

The linear-scaling DFT methods can be utilized to improve the efficiency of
DFT calculations. Linear-scaling DFT reduces the computational complexity of
DFT from $O(N^{3})$ to $O(N)$ by making use of the nearsightedness principle
[100]; therefore, it is highly advantageous for simulating HEAs, where
supercells with more than 100 atoms are commonly needed. Different linear-
scaling DFT methods have been proposed and implemented [101]. For example, the
locally self-consistent multiple-scattering (LSMS) method [102, 103] is
employed by Liu et al. [104, 105, 68, 69] to accelerate the calculation of HEA
dataset. The linear scaling in LSMS is achieved by restricting the quantum
scattering of electrons within the so-called local interaction zone, but the
long-range electrostatic interactions are still evaluated everywhere.

####  2.3.2 Data for physical-property models

There are various sources for the training data of ML models that predict bulk
physical properties. These data can be experimental data or simulation data.
Aided by powerful supercomputers and efficient algorithms, the volume of
simulation data grows quickly. In particular, the well-structured simulation
data are obtained with controllable and transparent conditions, which are
easily repeatable and findable. These advantages render them attractive as
fuel to power ML models.

Experimental database Pauling File is an excellent experiment database [106].
The Pauling File project was launched in 1995 with the aim to create tools for
scientists working with inorganic compounds, with a particular focus on
materials design. Another important one is the Materials Experiment and
Analysis Database (MEAD) [107]. In addition, some websites also provide
collected experimental data for pure elements, such as periodictable.com. Pei
et al. have used the data to train several ML models with good performance
[50, 48, 108].

Computational database There are multiple well-acknowledged DFT databases for
materials worldwide. Some well-known examples include the Materials Project of
Lawrence Berkeley National Laboratory, Open Quantum Mechanics Database (OQMD)
of Northwestern University by Wolverton et al. [109], and AFlow by Curtarolo
et al. at Duke University [110] in the U.S.A. In Europe, NOMAD is a
representative one. Draxl and Scheffler reviewed some recent progress in big-
data-driven materials science, particularly their effort to build the so-
called FAIR data infrastructure [111]. Another European DFT database is the
Automated Interactive Infrastructure and Database for Computational Science
(AiiDA) which can be accessed by its interactive platform, Materials cloud
[112]. Himanen et al. provided a good summary of the databases [113].

There are also published data sources specifically for HEAs. For example, in
2018, two databases of HEAs were compiled by Senkov and Miracle et al., with
one focus on the mechanical properties of 122 refractory HEAs [114], and the
other one on 370 HEAs [115]. These databases were updated in 2020 by Borg et
al. [116], with 1545 records of mechanical properties and phases from 265
articles being compiled. Gao et al. also published 1252 data of multicomponent
solid solutions and intermetallics [117], which was employed by Pei et al. to
train an ML model for phase prediction [48].

####  2.3.3 Data augmentation

Despite the rapid development in methodology and computing resources, DFT
calculations are still computationally expensive and inefficient. First, the
typical dataset size is limited to tens of thousands, beyond which the
computing cost would be too high. The whole strategy of replacing DFT with an
ML surrogate model would lose its merit. By comparison, in image recognition,
the simple MNIST handwritten digits database already has 70,000 instances,
while the popular ImageNet database for visual object recognition contains
more than 14 million instances [118, 119]. Secondly, the atomic configuration
space is enormous for multicomponent alloys [120]. For example, for a ternary
system in a 1000-atom supercell, the total number of configurations is
$3^{1000}$, which is much larger than the $3^{361}$ possible board
configurations in the game played by the famous alphaGo [121].

Active learning is commonly employed to iteratively improve the dataset
quality by incorporating important data points from simulations [122, 123]. In
each iteration of active learning, a query algorithm is employed to determine
the new data points, which are evaluated and added to the original dataset.
There are different strategies for choosing the data points. For instance, one
strategy is to recalculate the energies with DFT if the energies predicted by
the MC simulations are outside the energy range of the original dataset [69].
Similarly, in MD simulations, the trajectory-based iterative sampling
algorithms [124, 123] can be used to sample the dynamically important regions
in the configuration space and add new data points where the trajectories
visit most frequently. Another strategy is to estimate the prediction error
with multiple models and add new data when the prediction error is larger than
a threshold value, which is demonstrated for neural networks [125, 126, 127]
and Gaussian process [128, 129]. Finally, Shapeev et al. proposed an active
learning strategy based on the so-called D-optimality criterion [122, 130],
which selects the training set by maximizing the determinant of the
information matrix. This method has been implemented in the MLIP package
[131].

Another method to expand the dataset is the generative adversarial network
(GAN). GAN is a deep learning architecture comprised of generative and
discriminative parts. In the training process, the generative model tries to
produce faked data similar to the training data, while the discriminative
model attempts to distinguish the true and faked data. A well-trained GAN
should capture the training dataset’s important latent variables, therefore
serving as a powerful tool for data augmentation. For example, in a recent
work, Lee et al. [132] employed GAN to augment the dataset of HEA properties
and demonstrate that such a technique can significantly improve the
performance of a deep neural network model for phase prediction with the test
accuracy enhanced from 84.75% to 93.17%.

###  2.4 Model selection

Commonly used descriptors for phase prediction include atomic number, atomic
size, the entropy of mixing, enthalpy of mixing, electronegativity, valence
electron concentration, elastic constants, melting temperature, and many
others. In general, the relevance of each feature to the labels is unknown
beforehand, so a sufficiently large pool of features is needed to avoid the
case that essential features are left out. However, this large set of features
introduces redundancy and the curse of dimensionality, and a feature selection
procedure is necessary to reduce the model complexity and the overfitting
risk.

There are three categories of feature selection techniques. The first one is
to use measures such as correlation coefficients, mutual information, or
similarity scores to evaluate the features’ independence and their correlation
to the target variable. The second method uses regularization techniques such
as LASSO regularization to penalize non-zero coefficients and, therefore,
automatically construct a sparse model. Finally, a computationally more
intensive method is to use different feature sets to train models in a brute-
force manner and then use a hold-out set to measure the model performance and
select the best set.

Other than the descriptors, the hyperparameters in a model can also be
selected to achieve the best performance. The selection of model
hyperparameters can be divided into two major categories. The first one is
based on information criteria, such as the Bayesian information criteria (BIC)
and Akaike information criteria (AIC) [133]. The second one is to use hold-out
samples (e.g., cross-validation) to measure the model performance and tune the
hyperparameters. For such a purpose, grid search and random search are the two
commonly used simple methods. The hyperparameters can also be tuned with more
involved techniques, including genetic algorithm [134, 71], evolution
strategies [135] [136, 137], gradient-based optimization, and Bayesian
optimization methods [96] such as Gaussian process and tree-structured Parzen
estimator (TPE) [138].

###  2.5 Model training

Once an ML algorithm is selected, the ML model is completely determined by the
optimal model parameters. The model parameters can be determined via the
minimization of the loss function. An important point to note is that while
the ML atomic interaction models are built to predict the atomic energy, in
the training process, the target variable is the total energy computed by DFT
methods. For an effective Hamiltonian, the total energy is the only target
variable, while for interatomic potential, the atomic forces also need to be
considered. Representing the total energies and atomic forces of the $k$-th
configuration as $E^{k}$ and $F_{i}^{k}$ respectively, we calculate the mean
squared error (MSE) loss function by

$\displaystyle{\rm{MSE}}=\frac{1}{N_{\rm{config}}}\sum_{k=1}^{N_{\rm{config}}}\left[\left(E^{k}_{\mathrm{ML}}-E^{k}_{\mathrm{DFT}}\right)^{2}+w_{F}^{k}\sum_{i=1}^{N_{k}}\left(F^{k}_{i\mathrm{ML}}-E^{k}_{i\mathrm{DFT}}\right)^{2}\right],$
(1)

where $N_{\rm{config}}$ is the number of configurations in the dataset,
${N_{k}}$ is the number of atoms in the supercell of the $k$-th configuration,
and $w_{F}$ is the weight factor of atomic forces, which can be adjusted,
e.g., according to the number of atoms in the supercell. Based on the ML
model, the optimization of the loss function can be achieved with different
algorithms, such as the ordinary linear regression, Newton’s method,
stochastic gradient descent (SGD), and Adam optimizer [139].

ML models are usually data-hungry. One interesting question is whether some
form of local energies can be calculated by DFT and used for the fitting
process instead of the total energies [140]. One obvious benefit of this
method is that the number of data points can be greatly expanded from
$N_{\rm{config}}$ to approximately $N_{\rm{config}}\times N^{k}$. This is
particularly attractive for HEAs, where a large supercell is needed to
represent the random configurations. The real space DFT methods, such as LSMS,
are indeed feasible to evaluate the local contributions by integrating the
energy density over the local Voronoi polyhedron. However, one problem with
this method is that the local energies of neighboring atoms are highly
correlated. For example, suppose there is a charge transfer between two
neighboring atoms. In that case, the energy increase of one atom leads to
decreasing the energy of the other, while the total energy change is much
smaller than each individual atomic energy. As a result, although the atomic
energy can be determined with a small relative error, it still gives rise to a
large relative error for the averaged total energy, which has a standard
deviation smaller than the local atomic energies.

###  2.6 Result analysis

After the ML models are trained, the next step is to evaluate their
performance. For such purposes, hold-out samples that do not appear in the
training dataset are usually used. For an objective and accurate evaluation,
the testing samples should be representative and uncorrelated to the training
samples; otherwise, it may give rise to a spuriously high test score. For
classification tasks, quantities such as recall, precision, area under the
curve (AUC), or receiver operating characteristic (ROC) curve are commonly
used to measure the model performance. For regression tasks, the root mean
square error (RMSE) and coefficient of determination ($R^{2}$ score) are
widely used.

While well-trained ML models can be very good at making predictions, in many
situations, the reasons for their effectiveness are also of great interest.
For example, ML can be used to gain physics insights from the solutions
provided by the model [50, 48, 70, 108]. Model interpretation is relatively
easier for some ML models than others. For example, in decision trees, the
importance of features can be evaluated by inspecting their effects on
reducing the Gini index. For less transparent models, techniques such as the
permutation feature importance can be employed to identify the most important
features [141]. The permutation feature importance is defined by the decrease
of the model performance when the selected feature’s values are randomly
shuffled. This is more challenging for neural networks. For simple neural
networks, the sensitivity of the model with respect to the input features can
be approximated by using the Taylor expansion to convert the nonlinear model
into a linear one [84].

Deep neural networks are generally difficult to interpret due to their
nonlinearity and high dimensions. One important technique to visualize high-
dimensional data is the t-distributed stochastic neighbor embedding (t-SNE)
algorithm [142]. The t-SNE algorithm maps high-dimensional data into two or
three-dimensional space where one data point keeps the similar data as its
neighbors as in the original space. For classification, the latent variables
obtained by t-SNE are visualized to infer the model’s performance. If data
with different labels are well separated in the reduced space, then the model
can differentiate them easily; otherwise, the model is difficult to
distinguish them [132].

##  3 Atomic interaction models

One important requirement for atomic interaction models is that they should be
independent of the system size; thus, the model trained from small systems can
be employed to investigate much larger ones. For such purpose, the atomic
energy $E_{i}$ is a crucial quantity, which represents the energy contribution
from an individual atom:

$\displaystyle E=\sum_{i=1}^{N_{\rm{atom}}}E_{i}({\bm{\sigma}}_{i})+E_{0},$
(2)

where
${\bm{\sigma}}_{i}=\\{\sigma_{i}^{0},\sigma_{i}^{1},\sigma_{i}^{2},\cdots\sigma_{i}^{N_{n}}\\}$
represents the atomic environment of the $i$-th atom, with
$\sigma_{i}^{0}=(z_{i},\mathbf{r}_{i})$ represents the atomic species and
position of the $i$-th atom, and $\sigma_{i}^{j\neq
0}=(z_{j},\mathbf{r}_{ij})$ represents the surrounding $N_{n}$ atoms within a
cutoff radius $r_{c}$. $E_{0}$ represents the long-ranged electrostatic term
that depends only on the chemical concentration, not atomic arrangement. As a
result, the problem of establishing a mapping between configuration and total
energy is reduced to mapping an atom’s local environment to its atomic energy,
as illustrated in Fig. 3. Note that when the atomic forces are also needed in
the atomic interaction model, they should be fitted simultaneously with the
local energies.

###  3.1 Symmetry-preserving descriptors

An encoding scheme that maintains the underlying symmetry needs to be devised
to represent the atomic environment with high fidelity. As a categorical
variable, the atomic species can be represented with one-hot encoding, element
embedding [143, 70], or simply assigning a unique weight for each atomic
species. A similar technique is used in constructing the atomic neighbor
density function [144]. The Cartesian coordinates can not be directly used as
features since squeezing the positions into one-dimensional vectors would
completely lose the underlying symmetry of the system. One simple scheme to
address this problem is to use the radial distances $r_{ij}$ of atom $i$ to
its neighboring atom $j$ as the input. An obvious drawback of this approach is
that it ignores the angular distribution, which can be mitigated by adding the
angle $\theta_{ijk}$ between the two lines formed by the relative positions
$\mathbf{r}_{ij}$ and $\mathbf{r}_{ik}$. A well-known example is the
Parrinello-Behler descriptor (also known as the symmetry function), which uses
the product of a Gaussian function and a cutoff function to represent the
radial arrangement of atoms. The function of $\cos(\theta_{ijk})$ is adopted
to describe the angular distributions [145, 146]. Other symmetry invariant
quantities can also be used to represent chemical configurations, such as
power spectrum and bispectrum [147, 144], and moment tensors [148, 131].
Another approach is to use physical observations to represent materials, such
as the X-ray diffraction patterns [85].

###  3.2 Machine-learning potentials

Interatomic potentials are functions that describe the dependence of the
potential energy on the atomic positions, which is crucial for molecular-
dynamics simulations. Popular traditional interatomic potentials include the
Lennard-Jones potential, embedded-atom method (EAM) [149], and reactive force
field (ReaxFF) [150]. Compared to these traditional potentials, ML potentials
have two main features: Firstly, ML potentials take flexible rather than fixed
function forms, and their accuracy can be systematically improved; Secondly,
ML potentials adopt a data-driven approach comprised of training, validation,
and tests, using dataset calculated by first-principles methods. As a result,
high-quality ML potentials can approach the accuracy of quantum mechanical
methods such as DFT and enable the description of systems that are too complex
for traditional potentials depending explicitly on a few parameters. In the
following, we will give a brief introduction to four commonly used ML
potentials, i.e., the neural network potential (NNP) [145, 146, 151, 152], the
Gaussian approximation potential (GAP) [147, 144], the spectral neighbor
analysis potential (SNAP) [153, 154], and the moment tensor potential (MTP)
[148, 131].

####  3.2.1 High-dimensional neural network potential (HDNNP)

Artificial neural networks (ANN) are widely employed to establish a nonlinear
mapping between the atomic environment and the atomic energy. Typically a
feed-forward multilayer perceptron (MLP) is used, comprising an input layer, a
few hidden layers, and an output layer. A simple, fully connected layer is

$\mathbf{Y}=\phi(\mathbf{WX}+\mathbf{b}),$ (3)

where $\mathbf{X}$ is the input vector, $\mathbf{b}$ is the bias vector,
$\mathbf{W}$ is the weight matrix, $\phi$ represents the activation function,
and $\mathbf{Y}$ is the output vector. The vector $\mathbf{Y}$ is also the
input for its next layer. In forward propagation, the atomic environment
encoding mentioned is fed into the networks, and weight matrices in the hidden
layers allow the network to learn atomic interactions at different
hierarchical levels. The output layer predicts the atomic energies, which are
summed over all atoms within the system to predict the total energy, as shown
in Fig. 3. Since the neural networks are applied to every atom, NNP is also
referred to as high-dimensional NNP (HDNNP) [146]. The predicted energies are
compared in the backward propagation with DFT results to calculate the loss
function. By employing optimization algorithms such as the stochastic gradient
descent (SGD), the parameters $\\{\mathbf{W},\mathbf{b}\\}$ are then updated
to reduce the expected loss. The forward and backward propagation is iterated
until the model converges to satisfactory accuracy. Note that neural networks
with different sets of $\\{\mathbf{W},\mathbf{b}\\}$ parameters are used to
represent the individual atomic species, but the atomic neural network for a
given atomic species is the same. Different atomic energies are attributed to
the different environments of atoms.

####  3.2.2 Gaussian approximation potential (GAP)

In GAP [147, 144, 155, 156], the total energy is predicted by the Gaussian
process regression (GPR) method that measures the “similarity” to the
reference of atomic environments [157]. Since the Gaussian process is
computationally expensive for large datasets, a sparse representation [158]
technique is employed to reduce the computational cost of the GPR from
$O(N^{3})$ to $O(N^{2}M)$, assuming the number of data points $N\gg M$. The
choice of parameters for the kernel function $K$ reflects the assumed
covariance between the random variables. They measure the similarity between
atomic environments ${\bm{\sigma}}_{i}$ and ${\bm{\sigma}}_{s}$. For example,
if a dot-product kernel is used, the GPR is reduced to Bayesian linear
regression. The atomic environment ${\bm{\sigma}}_{i}$ can be represented by
the so-called atomic neighbor density

$\displaystyle\rho_{i}(\mathbf{r})=\sum_{j}w_{Z_{i}}f_{\rm{cut}}(r_{ij})\exp(-|\mathbf{r}-\mathbf{r}_{ij}|^{2}/(2\sigma^{2}_{\rm{atom}})),$
(4)

where $w_{Z_{i}}$ is a unique weight factor assigned according to the element
of atom $i$, $\sigma_{\rm{atom}}$ is the smearing parameter,
$f_{\rm{cut}}(r_{ij})$ is the cutoff function to ensure interactions beyond
$r_{c}$ are ignored, and the summation of $j$ is over all atoms within the
cutoff radius. Using Eq. 4, the so-called smooth overlap of atomic positions
(SOAP) kernel can be obtained by integrating the squared overlap of the
neighbor densities of atom $i$ and $s$, over all possible 3D rotations [157].

$\tilde{K}({\bm{\sigma}}_{i},{\bm{\sigma}}_{s})=\int_{\hat{R}\in
SO_{3}}d\hat{R}\left|\int d{\bf r}\rho_{i}({\bf r})\rho_{s}(\hat{R}{\bf
r})\right|^{2}$ (5)

In practice, the 3D rotation is accomplished analytically by expanding the
atomic density in terms of spherical harmonics, of which the integration can
be expressed in terms of Clebsch-Gordan coefficients. The derivation details
are well presented in Ref. [157] and are very similar to the SNAP, as
discussed below. Once the matrix expression of the SOAP kernel is determined,
the GAP can be obtained by solving the regression problem with the Gaussian
process.

####  3.2.3 Spectral neighbor analysis potential (SNAP)

SNAP is closely related to GAP in that the central quantity is also the atomic
neighbor density in Eq. 4. For simplicity, it can be assumed that the density
of the neighboring atoms for atom $i$ is made of a sum of atoms within the
cutoff radius, as represented by the Dirac $\delta$ function,

$\displaystyle\rho_{i}({\mathbf{r}})=\sum_{j}^{r_{j}<r_{c}}\delta(\mathbf{r}_{i}-\mathbf{r}_{j}).$
(6)

Expand the density using the spherical harmonics as the basis set, the above
expression gives

$\rho_{i}({\bf r})=\sum_{lm}c^{i}_{lm}Y_{lm}({\bf\hat{r}}),$ (7)

with the expansion coefficients

$\displaystyle c^{i}_{lm}=\sum_{j}^{r_{j}<r_{c}}Y_{lm}({\bf\hat{r}}_{j}).$ (8)

The coefficients can be used to construct rotation-invariant descriptors
since, in group representation theory, the direct product of two irreducible
representations can be decomposed into a direct sum of irreducible
representations. The simplest one is the power spectrum

$\displaystyle{p}^{i}_{l}$
$\displaystyle=\sum^{l}_{m=-l}c^{i*}_{lm}c^{i}_{lm},$ (9)

which is rotationally invariant since it commutes with the angular momentum
operator. Similar to the power spectrum, more complex invariants such as
bispectrum can be constructed using the radial distribution function rather
than Dirac $\delta$ function and map the atomic neighbor density within
$r_{c}$ to a sphere in four-dimensional space [153]. The power spectrum and
bispectrum in four-dimensional sphere can then be used as descriptors for
various regression models, including the linear SNAP [153], quadratic SNAP
(qSNAP) [154], and spectral neural network potential [159].

####  3.2.4 Moment tensor potential (MTP)

MTP uses the contraction of moment tensors to construct invariant descriptors.
In MTP, the total energy is expressed as

$\displaystyle
E=E_{0}+\sum_{i=1}^{N_{\rm{atom}}}\sum_{\alpha}\xi_{\alpha}B_{\alpha}({\bm{\sigma}}_{i}),$
(10)

where $\xi_{\alpha}$ are the coefficients that need to be determined from the
dataset. The basis functions $B_{\alpha}$ are defined in terms of the moment
tensor descriptors:

$\displaystyle
M_{\mu,\nu}({\bm{\sigma}}_{i})=\sum_{j}f_{\mu}(r_{ij},z_{i},z_{j})\underbrace{\mathbf{r}_{ij}\otimes\ldots\otimes\mathbf{r}_{ij}}_{\text{$\nu$
times}}.$ (11)

The radial function $f_{\mu}(r_{ij},z_{i},z_{j})$ plays the role of
“selecting” shells of atoms [160], and can be expanded in terms of a set of
basis functions as

$f_{\mu}(r_{ij},z_{i},z_{j})=\sum_{\beta=1}^{N_{Q}}c^{(\beta)}_{\mu,z_{i},z_{j}}f_{\rm
cut}(r_{ij})\varphi^{(\beta)}(r_{ij}),$ (12)

which is made up of the cutoff function $f_{\rm cut}(r_{ij})$ and the
Chebyshev polynomials $\varphi_{\mu}(r_{ij})$. The tensor part
$\mathbf{r}_{ij}\otimes\ldots\otimes\mathbf{r}_{ij}$ is comprised of the outer
product of relative position vectors, which contains angular information of
the atomic environment. $M_{\mu,\nu}$ can be interpreted as the $\nu$-th
moments of inertia weighted by the $\mu$-th radial functions, and
$c_{\mu,z_{i},z_{j}}$ are parameters to be fitted from the data. The basis
functions $B_{\alpha}$ are obtained from the contraction of moments
$M_{\mu,\nu}$. The so-called level of moments can be defined in terms of $\mu$
and $\nu$ [160], then up to the level cutoff, all tensor moments are included
in the local energy model. The MTP parameters $\theta=\\{\xi,c\\}$ are
determined by minimizing the loss function for the training dataset.

###  3.3 Effective rigid-lattice Hamiltonian

Compared to molecular dynamics, atoms in a rigid-lattice model are assumed to
be located on lattice sites. During Monte-Carlo simulations, only the atomic
species are updated according to a policy satisfying detailed balance, while
the atomic positions are not relaxed. In a lattice model, the total energy can
be expanded in terms of the cluster interactions

$\displaystyle E=E_{0}+\sum_{\alpha}V_{\alpha}\phi_{\alpha}(\bm{\Sigma}),$
(13)

where $\bm{\Sigma}$ represents one of the many possible configurations of a
system, $\phi_{\alpha}$ represents the correlation function for the cluster
$\alpha$ [42, 37]. The correlation function is determined by (i) the different
types of point, pair, triplet clusters, etc, (ii) the dimension of the
cluster, and (iii) the atomic species. An excellent property of the
correlation functions is they are orthogonal to each other in terms of the
lattice sites and atomic species. $V_{\alpha}$ are the coefficients of cluster
$\alpha$ known as effective cluster interactions (ECIs). The ECIs can be
directly calculated with coherent-potential approximation (CPA) methods such
as the generalized perturbation method (GPM) or the embedded cluster method
(ECM), but the most popular approach is cluster expansion [41, 42, 161], which
calculates the ECIs by fitting the DFT energies of selected configurations.
Another effective atomic interaction model is the low-rank potential (LRP)
[162] proposed by Shapeev, which uses low-rank tensors in the MTP method as
the expansion basis.

Despite its huge success in the theoretical study of alloys, the conventional
cluster expansion has been challenged by HEAs due to a large number of
interactions and the computational cost to extract the interaction parameters
accurately [163, 164, 31]. A large number of interaction parameters is prone
to overfitting, rendering the conventional structure-inversion method to
determine ECIs risky. As a result, it is advantageous to adopt a data-driven
approach for a robust model (Fig. 3), assisted by techniques such as adding
regularization terms and selecting suitable models. One simple example is to
find the ECIs via regularized linear regression:

$\displaystyle\hat{\mathbf{V}}=\mathop{\arg\min}\limits_{\mathbf{V}}\;||\mathbf{E}-\bm{\phi
V}||^{2}_{2}+g(\bm{V}),$ (14)

where $g(\bm{V})$ is the regularization term. One commonly used scheme for it
is the ridge regularization, i.e.,

$\displaystyle g(\bm{V})=\lambda||\bm{V}||^{2}_{2},$ (15)

where the $\ell_{p}$-norm is defined as
$||x||_{p}=(\sum_{i}|z_{i}|^{p})^{1/p}$, and $\lambda$ is a regularization
parameter that can be chosen to set the strength of penalizing large ECI
values. Instead of choosing the $\lambda$ by hand, Bayesian ridge regression
can be employed to automatically optimize $\lambda$ along with other
parameters by maximizing the log marginal likelihood. Sometimes sparse
solutions of the least square problem are preferred, where the $\ell_{1}$-norm
LASSO (least absolute shrinkage and selection operator) regularization can be
employed

$\displaystyle g(\bm{V})=\lambda||\bm{V}||_{1}.$ (16)

The so-called group LASSO regularization is also introduced in Ref. [165] to
handle multicomponent systems. On the other hand, model selection can be
employed to reach an optimal balance between accuracy and robustness. For
example, Zhang et al. [68] proposed to use the Bayesian information criterion
(BIC) for feature selection of the HEA energy model,

$\textup{BIC}_{\textup{RSS}}=n_{d}\log\left(\frac{\textup{RSS}}{n_{d}}\right)+k\log(n_{d}),$
(17)

where RSS is the residual sum of squares, $n_{d}$ is the number of observed
data, and $k$ is the number of parameters in the model. The first term
represents the training error, the second term represents the model
complexity, and a model is selected when $\textup{BIC}_{\textup{RSS}}$ reaches
minima. Another extensively used model selection technique is the cross-
validation score, which uses hold-out validation data to estimate the
performance of different models. It is particularly useful when the dataset
size is large.

Another difficulty is that the linear assumption in Eq. 13 may not hold when
only low-order terms are kept, and higher-order interactions are discarded,
which is common for HEAs due to the $N^{m}$ scaling behavior of $N$-site
clusters in an $m$-component system. To address this problem, nonlinear models
such as kernel ridge regression, quadratic regression, and neural networks can
be employed to model effective atomic interactions. For example, Natarajan and
Van der Ven [166] developed a formalism that uses nonlinear models to
calculate atomic energy. The local environment is represented with the so-
called local correlation functions $G_{\alpha}^{i}$, and the total energy can
be written as [166]

$\displaystyle E=E_{0}+\sum_{i}E_{i}(G_{\alpha}^{i},G_{\beta}^{i},\cdots),$
(18)

To describe the nonlinear effects, a fully connected 3-layer neural network
consisting of 4, 4, and 2 nodes are used, respectively. The neural network
model is trained with 1000 randomly generated distinct configurations and
tested with 1346 distinct configurations. The model consistently outperforms
the linear model, especially when the number of features is small. For
example, for six local features, the test error of the NN model is 0.006
eV/atom, while the linear regression model gives an error of 0.01 eV/atom.

Figure 3: A schematic of a machine-learning model for energy calculations.
(left) The atomic energy $E_{i}$ depends on the local atomic environment
within a cutoff radius $r_{c}$. The different colors of the atoms represent
different atomic species. The Cartesian positions and atomic species are
converted into symmetry-invariant representations as inputs to ML atomic
model. (right) A neural network is used as an example.

##  4 Machine-learning accelerated atomistic simulations

This section summarizes the progress of using ML surrogate models to
accelerate the DFT-based atomistic simulations. These simulations can be
divided into three categories: Monte-Carlo simulations, molecular dynamics,
and hybrid MC/MD simulations. The main advantage of Monte-Carlo simulations is
that, in principle, they can cover the entire phase space. This capability is
crucial for studying CSRO and phase evolution at different temperatures. They
are interesting not only because they can generate directly applicable results
but also provide essential inputs for studying the temperature dependence of
mechanical behaviors [167]. One main limitation of Monte-Carlo methods is that
they cannot describe non-equilibrium kinetic processes, such as the atomic
motions under stress. On the other hand, molecular dynamics is good at
studying atomic motions but typically only simulates a small portion of the
phase space due to the limited timescale. Hybrid MC/MD combines both methods
by introducing MC swap steps to the MD simulations to increase the phase space
visited. This acceleration is advantageous in many applications, although
either MC or MD inevitably introduces some drawbacks to this hybrid method. If
not suitably addressed, these drawbacks can also lead to erroneous results.
This section will also discuss ML-potential-based simulations for HEAs, as
illustrated in Fig. 4. The important applications are summarized in Tab. 2.

Table 2: A summary of machine-learning applications in the atomistic
simulations for multicomponent alloys. A few examples are given for pure
metals to illustrate the methods.

Materials |  Methods |  Physical Quantities |  Refs
---|---|---|---
MoNbTaW |  LRP+MC |  Phase evolution |  [168]
VCoNi |  LRP+MC |  Phase evolution |  [169]
AlNbTiV |  LRP+MC |  Phase evolution |  [170]
NiCoFeCr |  LRP+MC |  Phase evolution |  [171]
MoNbTaW, MoNbTaWV, MoNbTaWTi |  Bayesian CE+MC |  Phase evolution |  [68, 69]
MoNbTaWV |  GAP+ MC/MD |  Defects, segregation |  [172]
CoFeNi |  MTP+MD |  Local lattice distortion, CSRO |  [173]
TiZrHfTa |  MTP+MD |  Phase evolution, elastic constants |  [174]
VZrNbHfTa |  HDNNP+MD |  Melt structure, radial distribution function |  [175]
MoNbTaW |  SNAP+ MC/MD |  Dislocation, GSFE, Peierls stress, stress-strain curve |  [65]
MoNbTaW |  MTP+ MC/MD |  CSRO on the mobility of dislocations |  [64]
MoNbTa |  MTP+MD |  Unstable SFE |  [176]
Al-Cu |  HDNNP+MD |  GSFE |  [177]
Fe |  GAP+ MD |  Dislocation, GSFE, Peierls stress |  [178, 179]
Mo, Ta, Nb, W |  SNAP+ MD |  GSFE, Peierls stress |  [180]

Acronyms in the table: low-rank potential (LRP), Bayesian cluster expansion
(CE), Gaussian approximation potential (GAP), moment tensor potential (MTP),
high-dimensional neural network potential (HDNNP), and spectral neighbor
analysis (SNAP); chemical short-range order (CSRO), stacking fault energies
(SFE), generalized stacking fault energies (GSFE); Monte Carlo (MC), molecular
dynamics (MD), and hybrid MC/MD.

Figure 4: Schematic of atomistic simulations with ML models and their
applications. (left) The scheme to train various ML models through active
learning. (middle and right) Various thermodynamic and mechanical properties
can be obtained from the simulations. The figures are reproduced from [69, 34,
65].

###  4.1 Monte-Carlo simulations

Most of the fascinating mechanical properties of HEAs can be traced to the
complex interplay of chemical order and disorder. At first-order
approximation, random HEAs can be seen as made up of perfect crystals, with
each site occupied randomly by one of the principal elements. In this
simplified picture, random HEAs bear many similarities to pure metals. The
first correction to this simple approximation is the local lattice distortion,
induced by each atom’s different chemical environments. Another important
correction is the short-range and long-range order in the sample. As HEAs cool
down from the melting temperature, atomic pairs with favorable bonding energy
tend to appear more frequently, which gives rise to various degrees of order.
These inhomogeneities impede the motion of dislocations, which can have a
profound impact on strength and ductility. Therefore investigating the
occurrence of order-disorder transition can provide essential insights into
the exceptional mechanical properties of HEAs.

Driven by such motivation, Liu et al. [69] investigated three refractory HEAs:
MoNbTaW, MoNbTaVW, and MoNbTaTiW using canonical MC simulations. The MC
simulations were informed by an effective pair interactions (EPI) model. A
data-driven approach was adopted for model selection, with the Bayesian
information criterion [68] employed. They found the method produces highly
accurate Hamiltonians that give error bars less than 0.1 mRy for all three
HEAs. Moreover, the effective Hamiltonian accuracy was also validated by the
total energies of 1000-atom MC configurations, which were calculated by the
linear-scaling LSMS method. This demonstrates the importance of incorporating
MC configurations in the training dataset through active learning.
Furthermore, for each of the three HEAs, two order-disorder transition
temperatures, i.e., $T_{1}$ and $T_{2}$ are identified from the computed
specific heats and SRO parameters, as shown in Fig. 5. Finally, the
calculations demonstrate that MoNbTaW has much lower order-disorder transition
temperatures than MoNbTaVW, as illustrated in the specific heats vs.
temperature plots in Fig. 5 (a). Therefore, it is expected that in the range
between 1000 K and 2000 K, MoNbTaW is primarily a solid solution, while
MoNbTaTiW should contain a large fraction of second-phase precipitates. In
other words, MoNbTaW is expected to be more ductile than MoNbTaVW in that
temperature range, which is confirmed by the experimental results shown in
Fig. 5 (b).

There are also many other studies that combine machine learning models and
Monte-Carlo simulations. In conjunction with rigid-lattice MC simulations, the
low-rank interatomic potential (LRP) has been applied to compute the
thermodynamics of several BCC and FCC multicomponent alloys. For example,
Kostituchenko et al. investigated the order-disorder transitions in MoNbTaW
[168]. The lattice distortion effect ignored in the rigid-lattice model was
included by using DFT-relaxed structures. They found that the lattice
relaxation is important for MoNbTaW when the temperature is below 500 K. Using
the same approach, Kostituchenko and Körmann et al. also investigated the
short-range order in BCC AlNbTiV [170] and FCC VCoNi [169]. For AlNbTaV, their
calculations reveal a B2 ordering below 1700 K, which is mainly caused by the
site preference of Al and Ti atoms. For VCoNi, they found an order-disorder
transition around 1500 K, which was believed to be caused by the strong
ordering tendency of Co-V and Ni-V pairs of the nearest neighbors. They also
found that due to the relatively large size mismatch, neglecting the lattice
relaxation can increase the predicted transition temperatures by more than
30%. LRP has also been applied to study the FCC NiCoFeCr HEA [171] by Shapeev
et al., where they found that Fe and Cr form sublattices at temperatures lower
than 600∘C and 1230∘C, respectively.

Figure 5: The chemical ordering in HEAs calculated by a machine-learning
potential. Its relationship to strength and ductility is also explored. (a)
Identification of order-disorder transitions in MoNbTaW and MoNbTaVW. (b)
Experimental measurement of peak strains vs. temperatures in MoNbTaW and
MoNbTaVW. (c) Illustration of the connections between different ordering and
strengthening mechanisms. Figs. (a) and (b) are reproduced from Ref [69].

###  4.2 Molecular-dynamics simulations

In addition to MC, molecular dynamics informed by ML models is also a common
method for studying multicomponent alloys. Jafary-Zadeh et al. [173]
investigated the local lattice distortion (LLD) effects in the CoFeNi base
alloy through MD simulations with a moment tensor potential (MTP). The sample
was melted at 2300 K for 10 ns, then quenched to 300 K, and kept in
equilibrium for 5 nanoseconds. Based on the configurations obtained from MD
simulations, they successfully separated the static, dynamic, thermal
expansion, and chemical short-range order (CSRO) contributions to the LLD, as
well as their impacts on the elastic properties. Gubaev et al. [174]
investigated the elastic constants of TiZrHfTa by chemically tuning the
concentration of Ta, using MTP-informed MD simulations. They found that
structural phase change can profoundly impact elastic properties. Other than
MTP, high dimensional neural network potential (HDNNP) has also been applied
to study the VZrNbHfTa melt [175]. The structure of this melt is calculated
with both ab initio molecular dynamics (AIMD) and HDNNP. The results from AIMD
and HDNNP were compared by analyzing the partial radial distribution
functions, and a good agreement between the two methods was found.
Furthermore, their simulation results show that vanadium atoms dislike other
atomic species. Such an effect reduces the probability of forming a single-
phase solid solution in this alloy.

###  4.3 Hybrid MC/MD simulations

The generalized stacking fault energy (GSFE) is a critical quantity to
describe plastic deformation behavior. It determines the core structure of
dislocations and the minimal stress required for the onset of dislocation
motions, i.e., Peierls stress. Extra energies are introduced when a perfect
crystal is divided into halves by a slip plane and shifted along the slip
direction of a slip system in question. The defected system is typically
represented by a large supercell to avoid the spurious interaction between the
faulted planes. A GSFE is calculated as the increased energy relative to the
perfect crystal. Many different configurations need to be evaluated to account
for the chemical disorder, further increasing the computational cost.
Therefore, it is highly desirable to use efficient ML potentials to accelerate
the GSFE calculations for multicomponent alloys.

Ong et al. investigated the strengthening mechanisms in MoNbTaW with hybrid
Monte-Carlo/molecular-dynamics (MC/MD) simulations informed by a SNAP model
(Fig. 6) [65]. The GSFEs are calculated with a large supercell containing
36,000 atoms, which is beyond the capability of conventional DFT. They found
that the GSFEs of MoNbTaW are closer to those of W and Mo but much larger than
those of Ta and Nb. They also determined the dislocation core structures by
directly inserting a dislocation into the supercell and relaxing the structure
with the SNAP model. The Peierls stress was then measured when the dislocation
started to move. The Peierls stresses’ values are 1620 $\pm$ 637 MPa for screw
dislocations and 320 $\pm$ 113 MPa for edge dislocations. Finally, they also
evaluated the uniaxial compressive stress-strain behavior of MoNbTaW
polycrystals, with segregation to the grain boundary and SRO distributions
obtained from MC/MD simulations. In addition, the MTP potential is also
employed to study the impact of CSROs on dislocation mobility in the
refractory HEA MoNbTaW, using a hybrid MC/MD simulation with a huge supercell
of 573,672 atoms [64].

Figure 6: Applications of ML potentials to simulate dislocation properties.
(a) Comparison of the energies and forces calculated from an ML model with
those from DFT. (b) Comparison of the GSFEs calculated from a SNAP potential
(lines) and DFT (square markers) for metals and NbMoTaW in special quasi-
random structures. (c) Displacement vs. time curve of a screw dislocation
under shear stress. The left subfigure corresponds to a random sample at 1.4
GPa and 1200 K, while the right one corresponds to an SRO sample at 2.0 GPa
and 1200 K. Figs. (a) and (c) are reproduced from Ref. [64] and (b) from Ref.
[65], under a Creative Commons CC BY license.

Using an MTP potential, Hodapp and Shapeev calculated the $1/4$[111] unstable
stacking fault energies (SFEs) of Mo-Nb-Ta alloys with varying chemical
concentrations [176]. They demonstrated that the unstable SFE predicted by MTP
is in excellent agreement with DFT, with a maximum relative error of
$6$-$7\%$. They also showed that the DFT training data could be reduced from
the order of 10,000 to about 50-100, assisted by active learning. This feature
renders the MTP method attractive for the high-throughput screening of
multicomponent alloys compared to the traditional DFT methods.

Curtin et al. employed HDNNP and kinetic MC simulations to study the
dislocations and GSFEs of several alloys, including Al-Cu [177], Mg alloy
[181], and Al-Mg-Si [182]. For Al-Cu, they calculated the GSFEs of the
$\theta$ and $\theta^{\prime\prime}$ precipitates using HDNNP, a type of
angular-dependent potential (ADP, an extension of EAM potential), and DFT. The
results demonstrated that the neural network potential better agrees with the
experimental results compared to the ADP EAM potential.

In a more recent study, Byggmästar et al. [172] developed a tabulated Gaussian
approximation potential for MoNbTaWV and used it to study the radiation damage
effects via hybrid MC/MD simulations. In the bulk alloy, they observed an
ordering of Mo-Ta and V-W, consistent with other studies [69]. They also found
that vanadium in the damaged alloys segregates to the compressed interstitial-
rich regions.

In addition to multicomponent alloys, ML potentials are also developed to
study the GSFEs of pure metals, which is more straightforward for comparison
with DFT methods. Dragoni et al. calculated the GSFEs in BCC iron with a GAP
model trained with 150,000 local atomic environments, and the results are in
excellent agreement with that of DFT [178]. The GAP model outperforms EAM
potentials. Maresca et al. further investigated dislocation cores in BCC iron
with the GAP model [179]. The Peierls stress was calculated to be about 2 GPa,
consistent with the DFT prediction. Using a SNAP potential, Wang et al.
studied the GSFEs and Peierls stresses in refractory BCC metals Mo, Ta, Nb,
and W [180]. Reasonable agreement with DFT was achieved for all the metals.

Figure 7: Machine-learning models for macroscopic properties. (left column)
The essential components of machine-learning models; (middle and right) The
thermodynamic and mechanical properties that can be calculated from the
models. For a better understanding of the physics of the problem, relations
between the descriptors are analyzed. The figures are reproduced from Ref.
[48] (phase prediction), [50] (descriptor analysis), [183] (elastic
constants), and [184] (yield strength).

##  5 Physical-property prediction

In this section, we will discuss the applications of ML algorithms to predict
physical properties. Important cases are summarized in Tab. 3 and Tab. 4. In
short, we will discuss the following aspects in this section:

Table 3: A summary of machine-learning applications for the predictions of
phases in HEAs.

Methods |  Target |  Dataset & Performance |  Refs |  Year
---|---|---|---|---
GP |  SS formation |  AUC = 0.97, 1252 data, |  [48] |  2020
GP |  SS formation |  >300 data |  [185] |  2017
KNN, SVM, NN |  Phase |  74.3% accuracy (NN), 401 data |  [186] |  2019
NN |  Phase |  over 80% accuracy, 118 data |  [187] |  2018
SVM |  Phase |  over 90% accuracy, 322 data |  [188] |  2019
DNN+GAN |  Phase |  93.17% accuracy, 989 data + GAN augmented data |  [132] |  2021
Model selection with GA |  Phase |  88.7% accuracy for solid solution formation and 91.3% for crystal structure |  [134] |  2020
Gradient-boosting |  Phase and Young’s modulus |  61% accuracy for phase, 0.1% to 42% error for Young’s modulus, 329 data |  [189] |  2020
Gradient-boosted DTs |  Phase |  96.41% for predicting single-phase solid solution, 1807 data |  [190] |  2022
Random forest |  Phase |  100% training accuracy, 134 data |  [83] |  2020
NN |  Phase |  Pearson’s R=0.983, 321 data |  [191] |  2020

Acronyms in the table: Gaussian process (GP), K-nearest neighbors (KNN),
neural network (NN), support vector machine (SVM), deep neural network (DNN),
generative adversarial networks (GAN), genetic algorithm (GA), decision trees
(DTs), the area under the curve (AUC); Solid solution (SS).

  * •

Use ML algorithms to predict solid solution, intermetallics, amorphous, and
mixed phases [134, 186, 187, 185] [48], as shown in Fig. 7.

  * •

Use ML algorithms to predict the crystal structure, such as FCC, BCC, HCP [83,
188, 132].

  * •

Identify important descriptors and design new prediction rules with the help
of ML [84] [50, 48].

  * •

Use ML algorithms to predict the mechanical properties of HEAs [71, 192, 193].

###  5.1 Prediction of solid solutions and multi-phases

Thermodynamics determines the phases in alloys, which provides essential
information to design high-performance alloys, including HEAs. Initially, HEAs
were defined as equiatomic single-phase multicomponent alloys; therefore, many
pioneering HEA studies focused on phase formations. The published
thermodynamic data renders the ML predictions for phase formation possible,
which is an active research area at the boundary of ML and HEAs. Below we will
present some representative examples and briefly summarize a number of other
studies.

Usually, we need to predict the phases for a given alloy composition. Its
inverse problem, i.e., identifying the alloy compositions that meet specific
targets [194], is even more interesting. The targets can be thermodynamic,
mechanical, or other physical properties. The vast compositional space of HEAs
is largely unexplored, and the synthesized HEAs are only a tiny fraction of
the literally uncountable HEAs. This feature renders an efficient exploration
of the HEA composition-phase space particularly interesting. Actually, there
are some exciting proceedings in this direction [195]. The inverse problem for
alloy design is mathematically equivalent to a constraint satisfaction problem
that can be solved using more efficient ML techniques, e.g., supportive vector
domain description. However, there is still a lack of reliable and efficient
methods to inform the algorithms and validate their predictions. The
CALculation of PHase Diagram (CALPHAD) method can be a good choice when
experimental thermodynamic data become increasingly available to inform and
validate the CALPHAD method. Another choice would be ab initio thermodynamics,
given that new efficient algorithms are constantly proposed and powerful
supercomputers have become increasingly more affordable.

High-quality (the most relevant) features and suitable ML algorithms are the
preconditions for the success of ML models in phase prediction in HEAs. Many
descriptors can be used as features for ML models. Zhang et al. performed a
comprehensive study that involved many descriptors and ML algorithms, making a
total of 45 ML models [134]. Using genetic algorithms (GA), they selected the
best ML model and material descriptors from a large model pool and
demonstrated their efficiency on two-phase formation problems in HEAs. The
optimized model demonstrates an accuracy of 88.7% for identifying solid-
solution and non-solid-solution phases and an accuracy of 91.3% in predicting
body-centered-cubic (BCC), face-centered-cubic (FCC), and dual-phase HEAs.
Similarly, Huang et al. also predicted the phase formations in HEAs [186,
187]. They compared the accuracy of the K-nearest neighbors (KNN), support
vector machine (SVM), and artificial neural network models on the prediction
of the solid solution (SS), intermetallic (IM) compound, and mixed SS and IM
phases. They used an experimental dataset of 401 HEAs, including 174 SS, 54
IM, and 173 SS+IM phases. The ANN model is the most accurate, with a value of
74.3%, compared to 68.6% and 64.3% for KNN and SVM.

Identifying high-entropy solid solutions is one of the primary research
interests among the HEA community. There are a large number of ML studies on
this topic, which render it difficult to summarize all of them exhaustively.
Here we can only briefly summarize several of them. Based on a critical
assessment and a Gaussian process statistical analysis, Tancret et al.
proposed a robust strategy to predict the formation of solid solutions [185].
They took into account most of the previously proposed criteria
simultaneously. The method can be readily used as a guide to design new solid-
solution HEAs. Lee et al. predicted AM, SS, IM, and mixed IM and SS phases
with a deep neural network [132]. The hyper-parameters for the regularized
deep neural network are searched via Bayesian optimization[96]. To obtain the
large dataset needed for neural network models, they employed conditional
generative adversarial networks (GAN) to generate additional data (data
augmentation), which can improve the performance of the neural network model.
They also demonstrated that the optimized neural network model could reach an
accuracy of 84.75%, performing superior to SVM, decision tree, and XGBoost
models. Roy et al. predicted Young’s modulus, as well as phase identification,
for a range of multicomponent alloys [189]. They use gradient-boosting
decision trees to predict the phases and Young’s modulus. The agreement is
generally good for dozens of multicomponent alloys in the Mo-Ta-Ti-W-Zr
family. Kaufmann et al. also evaluated the performance of random forests for
predicting solid-solution formations [83].

A finer level of the problem is to predict the crystal structures of the
formed solid solutions. Li et al. [188] use support vector machine model to
distinguish different crystal structures. The trained model is employed to
make predictions of the crystal structure for a large compositional space made
up of 16 elements as an effort to design new alloys. A summary of the above
examples is given in Tab. 3.

###  5.2 Machine-learning informed new rules for phase predictions

In addition to predicting phase formation, ML can also offer physical insights
into the role of elements in phase formation. For example, in one study,
researchers uncovered the eutectics design by ML in the Al-Co-Cr-Fe-Ni high-
entropy system. ANN models classified the roles of elements [191]. In the Al-
Co-Cr-Fe-Ni system, a database of 321 alloys was built up using the literature
results and CALPHAD calculations. It contains compositions and phase
constitutions of these alloys. The ANN model was then trained to predict many
near-eutectic compositions. The findings from the AI model can be used to
formulate a new rule of thumb for designing high entropy alloys.

Figure 8: The machine-learning informed prediction of solid solutions. (a) The
correlation matrix of elemental properties and alloy phase. The matrix shows
that simple pair correlation is not enough to capture the mapping from
elemental properties to alloy phases well. (b) The receiver operating
characteristic (ROC) of Gaussian process classification with confidence band
for single-phase versus multi-phase. (c) The receiver operating characteristic
(ROC) of Gaussian process classification with confidence band for face-
centered cubic (FCC) versus body-centered cubic (BCC) versus hexagonal closest
packed (HCP) single-phase classification, respectively. The area under the
curve (AUC) is calculated for each ROC curve. (d) Gaussian process
classification (GPC) probability as a single-phase alloy versus atomic size
difference. The symbols (triangle, square, hexagon, and circle) represent the
experimentally measured phase states. (e) Prediction of the new rule
$\gamma\geq 1$ the new rule alone gives an accuracy of 73%, but with the
lattice misfit rule, it slightly increases to 81%. For better visualization,
the values of $\gamma>3$ are shifted to 3. (f) The same experimental data
points cannot be well classified by previously proposed rules. Figures (a)-(e)
are reproduced from Ref. [48], and (f) is reproduced from Ref.[117].

We provide a detailed example to elaborate our discussion on the physical
insights that ML can offer. Specifically, we demonstrate how to use ML to
improve the prediction of thermodynamic properties of HEAs. Different from all
above-mentioned examples, this example goes beyond the traditional ML
framework, and demonstrates how ML can help construct a new thermodynamics-
based rule to predict the formation of high-entropy solid solutions (Fig. 8)
[48]. We adopt the Gaussian Process Classification (GPC) algorithm for this
purpose. GPC algorithm is a Bayesian method that constructs a distribution
based on existing data. Since it provides distribution for each prediction
rather than a number, it can mitigate overfitting, a common problem for ML.
When the data volume is large, it yields a distribution with a small variance;
when the data volume is small, the variance is large. This feature renders it
suitable for a small dataset size, which is usually the case for materials
science problems. We collected 1252 multicomponent and binary alloys whose
phases (BCC, FCC, HCP, or multiphase) are known from the literature. Many of
these alloys have been described by Gao et al. [117]. Alloys with single
phases are labeled as "1", whether BCC, FCC, or HCP. The multiphase alloys are
labeled as "0". The features are constructed based on the 85 elemental
properties available for relevant elements in the Periodic Table of Elements
[196]. The physical properties of each multicomponent alloy are taken as the
averaged elemental properties over their concentrations. The properties are
brutal-force correlated with the phase states, and some important ones are
shown in Fig. 8(a). The performance of our GPC model is measured by the so-
called area under the curve (AUC) measure (see 8b-c). The model’s overall
performance yields a very high AUC of 0.97; more detailed performance for each
involved crystal structure is close, which is 0.96-0.97, showing the
performance is not biased among the different crystal structures.

The model performance is more clearly seen by its accuracy of prediction. The
criterion is that an alloy with a probability larger than 0.5 is considered to
be a solid solution. Half of the 1252 alloys are randomly picked for model
training and the others for validation with an accuracy of 93%. The
predictions are shown in Fig. 8(d) with the GPC criterion as one of the axes.
Almost all solid solutions, either FCC, BCC, or HCP, all have a probability
larger than 0.5, in contrast to the multiphase alloys. For better
visualization, the quantity of the commonly used lattice distortion criterion
is added. Alloys with lattice distortion $\delta\leq 6\%$ are generally
considered to form solid solutions.

The ML results significantly improve over the traditional criterion based on
the enthalpy of formation $\Delta H$. The enthalpy of formation measures the
thermodynamic tendency of the atoms to bind together. Too positive $\Delta H$
cannot bind atoms together (phase separation), while too negative $\Delta H$
will form intermetallics rather than a solid solution. The favorable energy
window is $\Delta H\in[-16.25,5]$ kJ/mol for solid solutions. However, this
empirical rule is inaccurate since many multiphase alloys also have formation
energy in this energy window, as is shown in Fig. 8(f). Compared to the
empirical rule, the machine-learning model is indeed very encouraging.

As demonstrated by the ML model, the solid solutions can be accurately
determined by the set of features that define a complex high-dimensional
space. The inaccuracy of the one-dimensional empirical rule must originate
from the inappropriate projection direction of the high-dimensional space. The
next question would be, can we construct a more accurate rule informed by the
ML results and the thermodynamic laws? We show this is indeed possible. We do
not offer the details here but show the thermodynamics-based rule we derived
in Ref. [48]. With the most relevant physical properties identified by ML, we
find solid solutions have a so-called $\gamma\geq 1$, with $\gamma$ defined as

$\gamma:=\begin{cases}\Delta G_{N}/\min(\Delta G_{2})&\text{if~{}}\min(\Delta
G_{2})<0;\\\ -\Delta G_{N}/\min(\Delta G_{2})&\text{if~{}}\Delta
G_{N}<0\text{~{}and~{}}\min(\Delta G_{2})>0.\end{cases}$

$\Delta G$ is the free energy of multicomponent and binary alloys. They can be
analytically calculated, and their definition is referred to Ref. [48]. The
rule $\gamma\geq 1$ together with the lattice distortion $\delta$ parameter is
shown in Fig. 8 (e). The overall accuracy of 75%. The machine-learning
informed new rule is nonetheless less accurate but physically more transparent
and convenient for high-throughput screening in new alloy design.

###  5.3 Machine-learning based order parameters

Order parameters are used to describe the phase states of materials. It is one
of the central concepts of condensed matter physics and materials science
[197, 198, 199]. ML can provide a comprehensive scalar order parameter that
has its unique advantages [70]. This can also be deemed as one of the
contributions of ML to materials physics.

We elaborate on this idea by taking one of our recent studies as an example
(see Fig. 9). The neural network architecture is constructed based on the
variational autoencoder (VAE) and shown in Fig. 9(a). Both the encoder and
decoder consist of several composite layers, except the last convolutional
layer next to the output. Each composite layer consists of a 3D convolutional
layer followed by an average pooling layer. Embedded in two dense layers is
the latent variable $z$ that is sampled from a Gaussian distribution with a
mean $\mu$ and a standard deviation $\sigma$. The input is a cubic lattice
(small dots) labeled by atomic species on BCC lattice sites (filled circle).
Each atom species is placed into a separate channel. The illustration is for a
four-component Mo-Nb-Ta-W system with four channels corresponding to Mo, Nb,
Ta, and W (from top to down). The output is a reconstructed sample from the
latent variable $z$. Fig. 9(b) shows the 2D t-SNE plot of VAE 12D embedding on
the test data for MoNbTaW. Various phases (colored by temperatures) are
clearly separated in the 2D space. This indicates an order parameter can be
defined in this space based on a distance metric. Such an order parameter is
defined using the Manhattan distance, i.e., $Z^{op}$ (red, Eq. 19),

$\langle Z^{op}\rangle_{T}=\frac{1}{M}\sum_{X_{j}\in
T}^{M}\sum_{i}^{d}|z_{i}(X_{j})|,$ (19)

The equation can be used to calculate the VAE-based order parameter at various
temperatures. The inserted snapshots are representative configurations at
three different temperatures, showing the microstructural changes from (1)
strong B2 order, (2) partial B2 order, to (3) A2 (BCC) as the temperature
increases. The corresponding data points in the latent space are also marked
on the $Z^{op}$ vs. $T$ curve.

As a physical requirement, the second-order moment of the order parameter that
has a physical meaning of susceptibility should have a peak at the phase
transition point. Similar to the classical susceptibility, the VAE
susceptibility is calculated from $Z^{op}$,

$\chi(Z^{op})=\bigg{(}\langle(Z^{op})^{2}\rangle_{T}-(\langle
Z^{op}\rangle_{T})^{2}\bigg{)}\frac{N}{T}.$ (20)

The $\chi(Z^{op})$ for several HEAs are shown in Fig. 9(c). We compare the VAE
susceptibility with experimental data. As indicated by the shadow bands (with
arrows indicating an estimated range from experiments), it indeed correctly
signals the phase transition temperatures for the HEAs.

The performance of the VAE order parameter is demonstrated through direct
comparison with the classic SRO and LRO parameters (Fig. 9(d)). The LRO
parameter for each element and the total order parameters $\eta$’s in terms of
LRO is shown in the left panel; the right panel shows the SRO for each element
pair and total order parameters in terms of SRO. The overall trend reflected
by the classic order parameters, i.e., increase or decrease of the ordering
degree, is well captured by the new order parameter. Interestingly, the VAE
order parameter is highly consistent with the LRO parameter in the high-
temperature region; in contrast, it is more consistent with SRO parameters in
the low-temperature area. This demonstrates that the new order parameter is
not simply a surrogate of the classic order parameters. Instead, it has its
unique features that merit further investigations.

Figure 9: Neural network-based order parameter. (a) The neural network
architecture of the VAE model. Both the encoder and decoder consist of several
combinations of a 3D convolutional layer and a pooling layer, except the last
convolutional layer for output. The latent variable $z$ is sampled from a
Gaussian distribution with mean $\mu$ and standard deviation $\sigma$. (b) The
2D t-SNE plot of VAE 12D embedding on the test data. $t_{1}$ and $t_{2}$ are
t-SNE embedding variables transformed from the 12D latent variable $z$. The
inserted snapshots are sampled configurations at three different temperatures.
(c) Comparison of the VAE order parameter with experimental data. The
experimental data validate the performance of the VAE order parameter. (d)
Comparison between the VAE order parameter and conventional order parameters.
(left) The LRO for each element and the total order parameters in terms of
LRO. (right) The SRO for each element pair and total order parameters in terms
of SRO. This figure is reproduced from Ref. [70].

###  5.4 Mechanical-property prediction

Here we will discuss the applications of ML algorithms to predict the
mechanical properties, as illustrated in Fig. 7. The important cases are
summarized in Tab. 4.

Table 4: A summary of the machine-learning applications for predicting the
mechanical properties of alloys.

Methods |  Target |  Performance |  Refs |  Year
---|---|---|---|---
GP classification |  Ductility enhancing elements |  92 % consistency with the YSI descriptor for 76 chemical elements |  [50] |  2019
CCA+GA |  Hardness (HV) |  5/7 of the synthesized HEAs within the 90% prediction interval, 82 training data |  [71] |  2019
Linear and polynomial regression, SVM with different kernels, CART, NN, KNN |  Hardness (HV) |  RMSE of about 50 HV, 155 measured hardness data |  [192] |  2019
Linear and Bayesian ridge regression, RF, KNN, and SVM |  Yield strength |  $R^{2}>0.95$ for RF, up to 44 data for each temperature subsets |  [193] |  2020
SVM |  Hardness |  Five descriptors are identified as key features to the hardness; Recommended HEAs are synthesized and demonstrated very high hardness |  [200] |  2022
Deep sets |  Elastic Tensors |  ML models trained from elastic property dataset of 7086 cubic quaternary HEAs, calculated with EMTO-CPA |  [201] |  2022
RF |  Yield strength of MoNbTaTiW and HfMoNbTaTiZr |  2.5% to 7.7 % accuracy, 240 data |  [202] |  2021
NN |  $\gamma^{\prime}$ phase volume fraction and yield strength |  For $\mathrm{Ni_{32}Co_{28}Fe_{28}Cr_{3}Al_{3}Ti_{6}}$ HEA, ML prediction (53% volume fraction and yield strength of 1.067 GPa) agrees with measurements (50.4% volume fraction and yield strength of 1.03 GPa) |  [184] |  2021
Gradient-boosted decision trees |  Elastic constants |  For $\mathrm{Al_{0.3}CoCrFeNi}$, 6% error for bulk modulus, 10% error for shear modulus, trained on 6,826 data |  [183] |  2019

Acronyms in the table: Gaussian process (GP), canonical correlation analysis
(CCA), SVM (support vector machine), classification and regression tree
(CART), neural network (NN), k-nearest neighbor (KNN), random forest (RF);
yttrium similarity index (YSI), root-mean-square error (RMSE).

Identifying reliable descriptors for mechanical properties by researchers is
usually time-consuming since this may involve understanding the deformation
mechanisms. In contrast, ML algorithms can identify the descriptors
efficiently, given sufficient data from either experiments or simulations.
Actually, if the data volume is substantial, ML models can even take all data,
and the descriptor selection process can be ignored. Probably due to the
limited available experimental data as the training data for ML, there are few
representative examples for HEAs, particularly the ML models for ductility.
Here we use a binary Mg alloy as the example [50]. The method shall be
straightforward to extend to multicomponent Mg alloys [203] and HEAs.

There are two widely acknowledged mechanisms to explain the enhanced ductility
in Mg alloys upon alloying with rare-earth elements [204, 205][206]. One is
based on the nucleation mechanism of the non-basal dislocations, whose
critical role is guaranteed by the von Mises criterion [205]. The other
explanation focuses on the mobility of the non-basal dislocation [206].
Interestingly, both mechanisms can explain well available experimental
observations. Moreover, the two descriptors associated with these mechanisms
also successfully predict alloys with enhanced ductility. The origin of this
correlated effect is revealed by a theoretical model [204]. Therefore, it is
of great significance if ML can reproduce the predictions of the two
descriptors. This will demonstrate ML is not just an efficient method but also
a reliable and accurate one.

We performed DFT screening on 21 hexagonal close-packed elements, and 5 of
those Mg alloys were synthesized and tensile tested [207]. The experiments
indeed confirmed the enhanced ductility in these alloys predicted by DFT.
Given the insufficient experimental data for training, we adopt the Gaussian
Process Classification (GPC) algorithm that can avoid overfitting and is
particularly suitable for small datasets. The GPC model takes the 21 DFT
predictions as the labels. To build an unbiased model, we consider all
available 85 elemental properties and brutal-force down-select the ones that
are most relevant to the ductility (Figs. 10(a)-(b)). After fine-tuning the
model parameters, we show the ML predictions are highly consistent with the
two descriptors [50]. As shown in Fig. 10(c), the primary area is pink,
indicating the consistency of ML results with the two descriptors/mechanisms.

Figure 10: The machine-learning informed prediction of mechanical properties.
Here the ductility of Mg alloys is taken as an example. (a) The whole process
of our data analysis. (b) The correlation graph for the relevant properties in
feature selection. An edge is formed between 2 nodes if Pearson’s correlation
between the two properties has a coefficient of $>$0.2 and edge thickness is
weighted by the correlation value. Different colors represent different
modularity classes based on each node’s degree, and each node’s size is scaled
by its degree. The screening process identifies the boxed properties. (c)
Comparison of the machine learning (ML) solution, YSI descriptor (Mechanism
I), and $-\delta_{X}$ (Mechanism II). The cyan blocks represent where the two
predictions disagree, the blue ones represent that no data is available, and
the pink ones represent where the two predictions agree. The figures are
reproduced from Ref. [50] with changes.

Reliable ML models with carefully selected ML algorithms are not just
mathematical black boxes but can also contribute to understanding physics in
materials science problems. For example, when the two descriptors are not
consistent in several predictions, ML as a third method can help pick the more
promising descriptor whose prediction is consistent with it. The mechanism
whose descriptors are in better agreement with ML is more likely the one
active during deformation. Also, we can use ML to discover new descriptors
that are able to reproduce the ML results, as descriptors usually do not need
computer programming and are simpler than ML in applications. We give one
example in Ref. [50] and show it is possible to rediscover the descriptor for
the mechanism I. These results were summarized in Ref. [108] for more details.

In addition to the above example, we briefly summarize several representative
studies for other mechanical properties. Hardness is easy to measure and
offers a quick means to estimate the mechanical properties of alloys. Given
this fact, it is the targeted property in several ML studies for HEAs.
Rickmann et al. [71] proposed to use the canonical correlation analysis (CCA)
and genetic algorithm to identify important features that contribute to the
hardness of HEAs. As a result, they identified promising HEAs and confirmed
their exceptional hardness in experiments. In a similar study, Wen et al.
designed HEAs with high hardness assisted by ML models [192]. First, they
constructed several ML models to estimate the hardness, including a linear
regression model, a polynomial regression model, support vector regression
models with different kernels (linear, polynomial, and a radial basis
function), a regression tree model (CART), a neural network model and a
k-nearest neighbor model. Then, guided by the models, they synthesized a few
alloys with a hardness 10% higher than the best alloy in the training dataset.

Peng et al. successfully predicted the yield strength for high-temperature
multicomponent steels using ML models [193]. Adopting the actual processing
conditions and information from CALPHAD, the researchers developed ML models
that are able to reproduce the existing experimental yield stresses. Using the
random forest regressor, Bhandari et al. [202] predicted the yield stresses of
MoNbTaTiW and HfMoNbTaTiZr at 800 ∘C, 1200 ∘C, and 1500 ∘C. The results were
compared with experiments, and good agreement was obtained. Moreover, Zheng et
al. [184] fed the data of compositions-microstructure-properties into an
artificial neural network to predict the $\gamma^{\prime}$ phase volume and
yield stress. Guided by the ML model, they explored the relevant compositional
space and designed a HEA with superior strength and ductility. The experiment
validated the excellent mechanical property of the HEA, which gives a yield
strength of 1.31 GPa and a tensile elongation of 15%.

Except for hardness and yield stress, there are also studies focusing on the
elastic constants, which are directly calculated by DFT. For example, Kim et
al. carried out detailed calculations of the elastic constants of an FCC HEA
AlCoCrFeNi, focusing on the effect of lattice distortion [183]. They found
that including the lattice distortion in DFT brings the elastic constants
closer to experiments. Moreover, the ML model trained on a large dataset of
inorganic structures also accurately predicts the HEA’s elastic properties.

##  6 Challenges and Outlook

###  6.1 Challenges

Despite ML’s advantages for HEAs, challenges still need to be addressed to
realize their potential fully. The first is model confidence, i.e.,
understanding when the model gives accurate prediction and when it may not.
Without such knowledge, it would be very risky to trust the prediction of an
ML model, as illustrated in some well-known “adversarial attack” examples
[208]. One standard tool for this problem is uncertainty quantification (UQ),
which attempts to estimate the model uncertainty via statistical methods. It
is also desirable to carry out an objective, and extensive benchmark of
various ML potential and models [59, 61, 60, 209] [210], similar to the field
of DFT [211]. So, some heuristic rules can be established on the model’s
performance and applicability.

Another challenge is to learn the underlying physics efficiently. As
previously emphasized, the arrangement of atoms in materials has various
symmetries, such as permutation, translation, rotation, and crystal
symmetries, and the movement of atoms follows physical laws. Therefore, only
specific points and lines represent realistic systems in the high-dimensional
feature space. Currently, the most popular approach is to use hand-crafted
symmetry-preserving descriptors, as introduced in section 3.1. Despite the
considerable success of these methods, an end-to-end scheme that automatically
captures physics-relevant descriptors on the 3D lattice is still highly
attractive due to its broader applicability. One such example is a graph
neural network (GNN), where the local neighboring atoms and bonds are
represented by the nodes and edges of a graph rather than a set of pre-
calculated descriptors. However, end-to-end models generally demand a larger
dataset to train due to their enhanced flexibility. For example, Ref.[209]
demonstrates that a series of GNN methods require one or two magnitudes larger
datasets to gain similar accuracy as descriptor methods. Therefore, addressing
this challenge requires further progress in both ML models and data sampling.

The third challenge is obtaining a high-quality dataset, which is difficult
for a few reasons. First, experimental and simulation data face the problem
that different research groups usually make measurements under other
conditions. Combining these data requires careful calibration. Moreover, both
experimental and simulation data are time-consuming to obtain. As a result,
the dataset size is generally small, as demonstrated in Tab. 3. The small-data
problem is even more severe for HEAs than other materials, considering their
vast design spaces. A series of methods to improve the dataset has been
discussed in section 2.3.3. However, these methods are generally still
developing and cannot completely solve the small-data problem. Considering the
central role of datasets for ML models and the increase in model complexity,
constructing high-quality datasets will be essential for the development of ML
for HEAs in the foreseeable future.

Datasets play a central role in the quality of ML models. This role will also
profoundly affects sharing of experimental data among the materials science
community. For example, in scientific literature, it is common only to report
materials that demonstrate the desired properties, while the failed trials are
ignored due to the assumption that they are of little interest to other
researchers. However, such an assumption is not valid for ML methods. The
negative results are also valuable for quality-critical data balance, as
exemplified by a study on inorganic-organic hybrid materials [212]. Therefore,
to facilitate the development of ML models, researchers should be encouraged
to publish negative results along with positive ones. Another critical point
is that, in many cases, only relatively small data is available. This fact is
common for many physical properties due to the cost of experimental
measurements. For small datasets, ML models, such as linear regression and
support vector machines, are expected to perform better than the more
complicated ones, such as deep neural networks, which can easily lead to
overfitting. Finally, it is also essential to report the measurement details
to facilitate the use of physical-property data in ML. This is because the
measured values of alloys’ properties significantly depend on conditions such
as the samples’ synthesis, heat treatment, and measurement methods. Therefore,
before feeding the measured values to ML models, it is vital to carry out
careful data cleaning and calibration to ensure that these data are on the
same footing. Without such a procedure, the quality of the property dataset
can be compromised, which strongly affects the performance of the ML models.

###  6.2 Uncertainty quantification

Figure 11: Uncertainty quantification (UQ). (a) The components of UQ in ML
models: uncertainty sources, uncertainty representations, and uncertainty
methods. Each component is comprised of multiple methods. (b) One typical
example for the UQ applications in HEA design by Ha et al. [72] after
adjustments. They proposed an evidence-based recommender system for HEAs based
on the Dempster-Shafer theory, which is a general framework for reasoning with
uncertainty. The two sub-figures show the probability density functions for
the HEA rankings in different test sets. The model performance is better when
the probability density is distributed toward 0.

Most of the time, ML algorithms are used as “black-box” methods; therefore, it
is essential to conduct UQ to determine the reliability of the predictions
[213]. The confidence of ML models merits particular attention for
applications in HEAs, where the dataset size is typically not large, but the
design spaces are huge [10]. One critical approach to overcome this challenge
is uncertainty quantification (UQ), which gives an estimate of the uncertainty
for each prediction in a rigorous way [214]. In practice, there are different
approaches to estimating the uncertainty of models [215], as illustrated in
Fig. 11. For example, Ha et al. [72] recently developed an evidence-based
material recommender system (ERS) for HEAs, which utilizes Dempster-Shafer
(DS) theory, a general framework for reasoning with uncertainty. First, the
ERS system rationally considers each dataset as a source of evidence. Then, it
combines the evidence to draw the final recommendation of HEAs, where the
uncertainty can be quantitatively assessed. Except for DS evidence theory,
there are also many other UQ methods, including fuzzy sets [216], interval
methods [217], and information theory [218], as well as probabilistic
approaches such as Bayesian methods [219] and probability boxes [220]. For
example, Zhang and Shields [221] proposed a unified Bayesian framework that
allows quantifying uncertainties resulting from small datasets using
multimodel inference and efficiently propagating the uncertainties through
optimal importance sampling. They further investigated the effect of prior
probabilities on the evaluation of uncertainties given limited datasets.
Moreover, sampling-based Monte-Carlo methods can also be employed for UQ.
These methods are straightforward to implement but typically computationally
intensive. Several modern Monte-Carlo methods [222], including multilevel
Monte-Carlo [223, 224, 225], multi-fidelity Monte-Carlo [226, 227] and
multimodel Monte-Carlo methods [221, 228, 229, 230], are developed to address
the shortcomings. A confidence interval on the model prediction can be drawn
by identifying and quantifying the sources of the uncertainty. This not only
allows users to understand the prediction reliability but also facilitates the
implementation of active learning [231, 232] by adding data in the region of
high uncertainty, which better fills the design space [233].

###  6.3 End-to-end (automatically learned) descriptors

A different class of methods that do not explicitly depend on the atomic
energy decomposition in Eq. 2 are convolutional neural network (CNN) based
models, such as the graph convolutional neural networks (GCNN) [234, 235, 236]
and three-dimensional (3D) CNN [237, 238]. In principle, CNN models can find
the important features automatically compared to descriptor-based models. The
atomic environment is learned with convolution filters to determine the total
energy. CNN models usually use a large number of parameters and thus are, in
general, more complex than descriptor models. Therefore, it requires more
training data [209]. The most popular CNN model for atomic interactions is
GCNN, which demonstrates excellent interpretability [235, 239] by maintaining
the concepts of atoms and chemical bonds. Compared to conventional CNN, GCNN
employs graph-structured data rather than 2D or 3D grids to represent the
arrangement of atoms in the system. In each convolutional layer, the “atom”
nodes update their values by convoluting with their surrounding atoms and
bonds and output a graph with the same size and structure as the input graph.
This process is very similar to the numerical renormalization group. After a
few convolutional layers, the feature vector represents the “renormalized”
atoms. The convolutional layers are then followed by pooling and fully
connected (FC) layers to determine the total energy. By replacing these
pooling and FC layers with a simple summation, it is easy to see that the
outputs of the last convolutional layer can be interpreted as local atomic
energies. Different GCNN models have been proposed, such as SchNet [240],
MEGNet [241], DimeNet [242], and GemNet [242], and their implementation can be
found in the Pytorch geometric package [243]. A benchmark of common GCNN
models in various bulk, 2D, and nano-structure materials are presented by Fung
et al. [209]. GCNN models have been widely used for predicting the physical
properties of both molecules [244, 245, 244] and crystals [235, 246, 247].
However, the adoption of GCNN for the atomistic simulations of complex
materials, particularly HEAs, has been rare. One reason is that while end-to-
end models generally have better representation capability, they are also
prone to overfitting and require a much more extensive training dataset to
achieve high prediction accuracy.

###  6.4 Machine-learning guided inverse materials design

Materials design is central to materials science, which can also benefit from
the introduction of ML techniques. The design of new materials requires
identifying the possible element mix to produce the targeted material
performance. In contrast to researching the mechanisms underlying the
properties of known materials, this is essentially an inverse design process
[248, 249, 250, 251] [252, 253, 194, 254]. The targeted material properties
are the consequences of many factors, including the alloy components and the
processing. Given this reduction feature, the same properties can result from
many different combinations of the factors. Due to this asymmetry of
information flow and the lack of constraints, materials design is indeed a
very challenging task, accompanied by many failures.

Edison’s trial-and-error method has long been a norm for the design of new
alloys for thousands of years. However, this method has been confronted with
the challenge of increasingly more components and the resulting complicated
hierarchical microstructure. Fortunately, we have accumulated many
experimental data that can be used in ML models to design complex alloys.
Moreover, new experimental methods and algorithms armed with modern, powerful
supercomputers result in theoretical databases to overcome the common problem
of insufficient data for ML models. We provide an example of using microscopy
images in the inverse design of complex alloys. Given the lack of typical
examples of high-entropy alloys, we adopt a model for steels with about 20
components. Albeit with less concentrated elements, the many components of the
steels indeed pose challenges as serious as high-entropy alloys, if not more
severe. The same method is applicable for high-entropy alloys.

We provided a neural network solution for the inverse material design based on
the knowledge obtained from microstructure images (Fig. 12a). The inputs are
the microstructure images and mechanical properties measured experimentally.
The images are collected using a scanning electron microscope (SEM) with some
post-processing. The training data is fed into a neural-network model that
consists of three sub-models, i.e., the encoder, the decoder, and a regression
model. Each of them has its unique function. Details are referred to Ref.
[194]. As one of the key results, two examples using Fe and Mn concentrations
as labels are shown in Fig. 12(b). The model can separate the steels with
different components and mechanical properties (here, yield stresses at room
temperature).

The performance of the model to differentiate the complex steels encourage us
to go one step further for the inverse design (Fig. 12(c)). The main focus
here is to show the main ideas. The regression model was originally a
predictor that predicts material properties based on alloy components and
compares them with the measured ones to optimize the model parameters. We copy
the structure and optimized parameters of the regression model to construct a
designer. The designer has a reversed information flow, albeit with the same
structure and model parameter. It allows us to explore and determine the
optimal microstructure of the alloys and then transform the microstructure
into alloy components. This actually realizes the challenging task of inverse
design. The inverse design method proposed here can also be readily applied to
other material systems, given sufficient microstructure images. It is expected
that many similar methods will emerge and provide new momentum for designing
new materials with lower costs and shorter periods.

Figure 12: The machine-learning informed prediction of mechanical properties
and inverse material design. (a) The neural-network model constructed based on
the variational autoencoder (VAE) model and the pre-processing of the training
data. (b) The kernel principal component analysis (kPCA) of the latent space
of the neural network model. (c) The inverse material designer based on the
regression branch of the model in (a). The figures are reproduced from Ref.
[50] with adjustments.

Another vital application of ML in materials science is the automatic analysis
of scientific images from technologies such as X-ray diffraction (XRD),
transmission electron microscope (TEM), scanning electron microscope (SEM),
and atom probe tomography (APT). The primary motivation for adopting ML models
rather than doing it by researchers is to enhance the analysis efficiency and
avoid possible artificial errors [255]. For example, Li et al. [256] employed
CNN to identify the L12-type ordered nanostructures in the APT data of an FCC
Al-Li-Mg system. This task is prohibitively tricky for researchers due to the
millions of data points. In another example, Kaufmann et al. [257] classified
images with two commonly used neural network architectures, i.e., ResNet50 and
Xception, for the autonomous identification of crystal structures based on
electron backscatter diffraction (EBSD) patterns. Each model has an overall
accuracy of >90%. Currently, this research direction is still in its infancy.
Ragone et al. employed a fully convolutional neural network (FCNN) to
determine atomic positions from the pixels’ intensities of STEM images. As a
result, they obtained a sufficiently high level of confidence [258]. Finally,
data augmentation techniques are employed by Ma et al. [259] to mitigate the
small data size problem by generating synthetic data of polycrystal images.
With the further improvement of the methods and the accumulation of
experimental data, we can expect that autonomous image analysis can be used to
build practically useful models that directly predict the mechanical
properties from the microstructure images with high accuracy.

##  7 Conclusions

The synergy of ML and materials science promotes the formation of a burgeoning
and exciting field with exponential growth. High-entropy alloys are
particularly interesting candidates to demonstrate the power of ML due to
their superior mechanical properties, vast compositional space, and complex
chemical interactions. The ML studies of high-entropy alloys can be grouped
into different categories depending on the specific motivations. On the one
hand, based on DFT datasets, ML methods can be used to construct atomic
interaction models (AIMs) to describe the complex interactions in
multicomponent alloys. The AIMs help to avoid the time-consuming first-
principles thermodynamics simulations. On the other hand, ML can also be used
to build models to predict bulk properties such as phase formations, crystal
structures, elastic constants, and yield strengths, starting from high-
throughput experimental or calculation data.

Atomistic simulations require high-fidelity predictions of atomic energies and
forces; therefore, retaining the underlying symmetry of atomic systems is
crucial. To this end, a series of ML AIMs have been proposed [61, 64],
including HDNNP, GAP, SNAP, MTP, LRP, Bayesian CE. These ML models have
enabled simulations with near DFT accuracy at length and time scales
inaccessible with conventional methods [63, 62]. The excellent combination of
efficiency and accuracy benefits HEAs, where large-scale simulations are
required to understand the origins of the exceptional mechanical properties in
HEAs. Indeed, ML AIMs have been successfully applied to investigate the
thermodynamics and mechanical properties of HEAs via MC or MD simulations, as
summarized in Tab. 2. For bulk property models (BPMs), various ML methods,
such as NN, SVM, GP, and CART, are employed. The datasets can be obtained from
experiments or theoretical calculations. One crucial procedure for building
BPMs is the selection of features and models. Techniques such as feature
analysis, model regularization, and out-of-sample validation are needed. The
huge compositional space and complex chemical interactions of HEAs render ML
models advantageous over empirical rules. ML models have demonstrated
excellent performance in predicting the bulk properties of HEAs, such as phase
formations and mechanical properties, as summarized in Tab. 3. Moreover, the
ML models have been applied to guide the design of HEAs, with exceptional
mechanical properties validated by experimental measurements [84, 184].

Finally, some promising research directions are discussed and illustrated with
examples, including uncertainty quantification of ML models, end-to-end
descriptors, and the ML-assisted inverse design of materials. These research
directions have attracted some researchers and merit more research efforts due
to their importance. Employing ML methods to study high-entropy alloys is
expected to continue expanding rapidly, and many opportunities await further
exploration.

##  8 Acknowledgements

The work of X.L. and Z.P. was supported by the U.S. Department of Energy,
Office of Science, Basic Energy Sciences, Materials Science and Engineering
Division. The work of J.Z. was supported by the U.S. Department of Energy,
Office of Science, Office of Advanced Scientific Computing Research, and the
Artificial Intelligence Initiative at the Oak Ridge National Laboratory
(ORNL).

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