Machine Learning for High-entropy Alloys: Progress, Challenges and Opportunities

Xianglin Liu; Jiaxin Zhang; Zongrui Pei

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Machine Learning for High-entropy Alloys: Progress, Challenges and Opportunities

Xianglin Liu, Jiaxin Zhang, Zongrui Pei

arXiv:2209.03173v1Sep 7, 2022cond-mat.mtrl-scicond-mat.dis-nn

TL;DR

High-entropy-alloy modeling faces difficulty constructing empirical atomic interaction models amid correlated atomic effects. The review surveys machine-learning approaches for atomic and bulk-property prediction, reporting near-DFT-accuracy simulations and strong bulk-property prediction performance.

Problem
Constructing empirical atomic interaction models is difficult because atomic effects can be highly correlated.
Method
The review covers machine-learning models for atomic interactions, including atomic-energy modeling and graph neural networks based on local neighboring atoms.
Results
The reviewed ML models enable simulations with near DFT accuracy and show excellent performance in predicting bulk properties such as phase formations.
Takeaways & Limitations
ML enables simulations at length and time scales inaccessible with conventional methods and supports prediction of bulk HEA properties.
Takeaways & Limitations
The review identifies uncertainty quantification as a remaining challenge, particularly where dataset size matters for model applications in HEAs.

Abstract

from arXiv · show

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.

1 Introduction

HEAs combine exceptional mechanical properties with a vast, tunable compositional space, but their chemical complexity makes modeling, simulation, and efficient design difficult. This review presents ML as a framework for addressing these microscopic and macroscopic challenges.

HEA motivation: HEAs mix multiple principal elements to form single-phase random solid solutions, expanding exploration beyond conventional alloy-composition corners.
HEA motivation: HEAs can combine strength and ductility while offering tunable chemical order, disorder, and heterogeneous microstructures.Some HEAs overcome the conventional strength–ductility tradeoff, and their order parameters affect dislocation structures and stacking-fault energies.
Modeling challenges: High-dimensional chemical interactions make empirical atomic-interaction models difficult to construct and can cause cluster expansion and EAM methods to overfit.For five elements, pair interactions across four coordination shells already produce 40 distinct interactions.
ML opportunity: Trial-and-error design is laborious in HEAs, motivating surrogate models and empirical rules for screening phase formation, strengthening, and ductility.
ML opportunity: ML has enabled a data-driven HEA research paradigm spanning atomistic simulations, physical-property prediction, hidden-physics extraction, and materials design.The review compares ML methods, surveys microscopic and macroscopic applications, and discusses remaining challenges and future opportunities.

2 How to utilize machine learning for high-entropy al-

A typical HEA ML application proceeds from identifying a suitable problem through algorithm and data selection, training, model selection, and result analysis. The workflow emphasizes well-defined objectives, appropriate data, and interpretation of learned parameters.

Workflow: ML is most effective for HEA problems with abundant high-dimensional data and well-defined objectives.
Workflow: The workflow begins by identifying an HEA application that can be formulated as an ML problem, then choosing an appropriate algorithm.
Workflow: Collected data are divided into training, validation, and test sets for model fitting, selection, and performance evaluation.
Workflow: Result analysis can inspect model parameters directly or use dimension-reduction techniques to extract underlying physical knowledge.
Workflow: Model parameters can be optimized with methods such as stochastic gradient descent, while grid search and Bayesian optimization support model selection.

2.1 Application identification

HEA ML applications include surrogate modeling, atomistic simulation, bulk-property prediction, and extraction of hidden physical structure. Atomic interaction models support transferable simulations, while bulk-property models directly predict macroscopic quantities.

Application types: HEA ML applications are broadly divided into atomistic simulations and macro-scale physical-property prediction.
Surrogate models: Atomic interaction models use local atomic arrangements to approximate first-principles behavior and can evaluate multiple physical quantities.These quantities include formation energies, elastic constants, stacking-fault energies, and phonon spectra.
Surrogate models: Bulk-property models directly predict target properties without requiring expensive simulations.
Surrogate models: Atomic interaction models generally require higher prediction fidelity because one nonphysical force can corrupt a subsequent molecular-dynamics trajectory.
Hidden physics: ML can extract hidden physics by reducing dimensions, identifying important features, clustering unlabeled data, or learning low-dimensional latent spaces with VAEs.
Hidden physics: A VAE and ML-informed Monte Carlo configurations were used to construct a chemical-configuration-based order parameter for HEA phase transitions.The example illustrates ML contributions to a central problem in condensed-matter physics and materials science.

2.2 Representative machine-learning algorithms

Representative ML algorithms differ in their modeling assumptions, data requirements, interpretability, computational cost, and suitability for regression or classification. The review describes SVMs, Gaussian processes, decision trees and ensembles, and neural networks as commonly used approaches.

Algorithm choice should match the task objective and dataset size because different ML algorithms have different strengths and shortcomings.
Neural network (NN): Neural networks use layered neurons that transform weighted inputs through activation functions and are commonly optimized by stochastic gradient descent.The multilayer perceptron is a simple feed-forward architecture; CNNs, RNNs, and transformers are other frequently used structures.
Loss functions measure prediction deviation from target values, with quadratic loss common for regression and cross entropy widely used for classification.
Support vector machine (SVM): SVMs classify data by maximizing separation between groups and can handle nonlinear boundaries through kernel methods.The linear decision plane is determined by the closest data points, while soft-margin and kernel extensions address nonseparable and nonlinear cases.
Gaussian process (GP): Gaussian processes provide probabilistic predictions and straightforward uncertainty quantification but become computationally expensive for datasets exceeding 10,000 points.Sparse techniques can reduce the cost by selecting a subset of the full dataset.
Classification and regression tree (CART): Decision trees split data recursively using feature criteria, while random forests and boosting improve prediction by combining multiple trees.Random forests use bootstrap aggregation and randomized feature selection; boosting sequentially improves predictor performance.

2.3 Data collection

ML training data for HEA models come from DFT calculations, experimental databases, and computational materials databases. Dataset quality depends on homogeneous methods, sufficient size, representative sampling, and strategies that expand data or target informative configurations.

A high-quality ML dataset should use consistent DFT methods, contain enough data to capture the physics, and represent configuration space evenly.Consistency requires matching exchange-correlation functionals, plane-wave cutoffs, and k-point meshes so energy differences reflect atomic configurations.
Linear-scaling DFT reduces computational complexity from O(N^3) to O(N), benefiting HEA simulations that require large supercells.LSMS achieves linear scaling by restricting electron scattering to a local interaction zone while evaluating long-range electrostatic interactions everywhere.
DFT remains computationally expensive for HEAs because typical datasets are limited and multicomponent alloys have enormous atomic configuration spaces.A ternary 1000-atom supercell has 3^1000 possible configurations; supercells with more than 100 atoms are commonly needed.
Active learning: Active learning expands datasets by iteratively selecting informative configurations using energy-range checks, trajectory sampling, prediction-error thresholds, or D-optimality.
Data augmentation: GAN-based augmentation can improve phase-prediction performance, increasing test accuracy from 84.75% in the reported example.

2.4 Model selection

Model selection for HEA property prediction involves choosing informative descriptors and tuning hyperparameters. Feature selection reduces redundancy, dimensionality, model complexity, and overfitting risk, while validation and optimization methods identify suitable settings.

Because descriptor relevance is unknown beforehand, models need a broad feature pool, but redundant features create a curse of dimensionality.
Feature selection uses correlation or mutual-information measures, regularization such as LASSO, or brute-force comparison with a hold-out set.
Hyperparameters can be selected using information criteria such as BIC and AIC or by measuring performance on hold-out samples.
Grid search and random search provide simple hyperparameter-tuning strategies, while genetic algorithms, evolution strategies, gradient methods, and Bayesian optimization offer more involved alternatives.

2.5 Model training

ML interaction models are trained by minimizing losses based on DFT total energies and, for interatomic potentials, atomic forces. Local-energy fitting can expand training data for HEAs, but correlated atomic contributions can make total-energy errors relatively large.

Effective Hamiltonians target total energies, whereas interatomic potentials additionally require atomic forces during training.
The mean squared error loss weights total-energy and force errors, with the force weight adjustable according to factors such as supercell atom count.
Loss minimization can use ordinary linear regression, Newton’s method, stochastic gradient descent, or the Adam optimizer.
Local-energy fitting can expand the effective number of data points from Nconfig to approximately Nconfig × N_k, which is attractive for HEA supercells.Real-space DFT methods such as LSMS can evaluate local contributions by integrating energy density over local Voronoi polyhedra.
Neighboring local energies are highly correlated, so small relative atomic-energy errors can produce large relative errors in averaged total energy.Charge transfer can increase one neighboring atom’s energy while decreasing the other, even when the total-energy change is much smaller.

2.6 Result analysis

Result analysis covers evaluating ML predictions, interpreting model behavior, and checking whether test performance reflects generalization. It emphasizes representative hold-out data, task-appropriate metrics, feature-importance methods, and visualization of high-dimensional representations.

Hold-out samples not used for training are typically employed to evaluate ML performance.
Representative, training-uncorrelated test samples are required to avoid spuriously high test scores.
Classification performance is commonly measured with recall, precision, AUC, or ROC curves, whereas regression uses RMSE and R2 score.
ML models can provide physics insights, but deep neural networks are difficult to interpret because of nonlinearity and high dimensionality.
Feature importance can be assessed through tree-based Gini reduction or permutation importance, which measures performance loss after shuffling a feature.
t-SNE maps high-dimensional data into two- or three-dimensional space, where label separation indicates easier classification.

3 Atomic interaction models

Atomic interaction models represent total energy through local atomic environments encoded with symmetry-preserving descriptors. ML potentials use flexible, data-driven mappings and can approach quantum-mechanical accuracy, while different architectures provide distinct computational trade-offs.

Atomic environments encode species and positions within a cutoff radius, while long-range electrostatic energy depends on chemical concentration.
Atomic interaction models decompose total energy into atomic contributions based on local environments, enabling models trained on small systems to address larger ones.
Cartesian coordinates require symmetry-aware representations; radial distances alone lose angular information, motivating angular, power-spectrum, bispectrum, and moment-tensor descriptors.
ML potentials use flexible functional forms and first-principles training, validation, and test datasets, allowing high-quality models to approach quantum-mechanical accuracy.
Neural network potential: Neural-network potentials sum per-atom energy predictions, fit them against DFT results, and update weights through iterative forward and backward propagation.
For six local features, the NN model reaches 0.006 eV/atom test error versus 0.01 eV/atom for linear regression.

4 Machine-learning accelerated atomistic simulations

ML potentials accelerate atomistic simulations by combining data-driven interaction models with MC, MD, and hybrid MC/MD methods. Applications span ordering, phase evolution, lattice distortion, defects, and dislocation behavior, with active learning improving accuracy and efficiency.

MC can in principle cover the entire phase space for studying chemical short-range order and phase evolution, whereas MD captures atomic motions but samples limited timescales.
Hybrid MC/MD adds MC swap steps to MD to increase visited phase space, but drawbacks from both methods can produce erroneous results if unaddressed.
Monte-Carlo simulations: Active-learning effective Hamiltonians for three refractory HEAs achieved error bars below 0.1 mRy and were validated on 1000-atom MC configurations.
Monte-Carlo simulations: The simulations identified two order-disorder transition temperatures for each refractory HEA using specific heats and short-range-order parameters.
Monte-Carlo simulations: MoNbTaW has lower order-disorder transition temperatures than MoNbTaVW, and experiments confirm the associated temperature-dependent ductility difference.
ML potentials reproduce DFT or experimental references for generalized stacking-fault energies, dislocation properties, local environments, and mechanical behavior across several alloys.
MTP predicts unstable stacking-fault energies within 6–7% maximum relative error while reducing DFT training data from about 10,000 configurations to 50–100 through active learning.

5 Physical-property prediction

Machine learning is applied to predict HEA phases, structures, descriptors, order parameters, and mechanical properties, while also extracting physical insights and supporting alloy design.

Phase and structure prediction: ML models predict solid-solution, intermetallic, amorphous, mixed phases, and FCC, BCC, or HCP crystal structures.
ML-informed rules and descriptors: ML identifies important descriptors and enables new thermodynamics-based rules for predicting high-entropy solid-solution formation.
Phase and structure prediction: 88.7% accuracy identifies solid-solution versus non-solid-solution phases, while 91.3% accuracy predicts BCC, FCC, and dual-phase HEAs.
Machine-learning-based order parameters: The VAE-based order parameter captures strong B2, partial B2, and A2 ordering as temperature increases and signals phase transitions.
Machine-learning-based order parameters: The VAE order parameter agrees with long-range ordering at high temperature and short-range ordering at low temperature, while retaining distinct features.
Mechanical-property prediction: ML models reproduce mechanical properties, identify promising hard alloys, and support experimentally validated designs with 1.31 GPa yield strength and 15% tensile elongation.

6 Challenges and Outlook

The review identifies limited data, model confidence, physics-aware representation, and benchmarking as central challenges for ML in HEAs, and highlights uncertainty quantification and inverse design as future directions.

Uncertainty quantification: Reliable model confidence is needed to determine when predictions are accurate and when they may not be, supporting safer application and active learning.
Physics-aware modeling: End-to-end models that automatically capture physics-relevant 3D-lattice descriptors remain attractive, but GNNs may require one or two magnitudes larger datasets than descriptor methods.
Data quality and scale: HEA datasets are generally small because experimental and simulation data are costly and require calibration across differing conditions.
Data quality and scale: Negative experimental results and detailed synthesis, heat-treatment, and measurement conditions are important for balanced, calibrated property datasets.
Uncertainty quantification: Uncertainty quantification estimates prediction uncertainty through approaches including Dempster-Shafer evidence theory, Bayesian methods, fuzzy sets, and interval methods.
Inverse materials design: Inverse-design studies use neural networks and microscopy images to distinguish complex steels and pursue alloys with targeted compositions and properties.

7 Conclusions

Machine learning addresses HEA complexity through atomic interaction models for large-scale simulations and bulk-property models for prediction and design. The review reports successful applications while identifying uncertainty quantification, end-to-end descriptors, and inverse design as continuing priorities.

7 Conclusions: ML studies of HEAs span atomic interaction models based on DFT datasets and bulk-property models for predicting composition-dependent properties.AIMs describe complex multicomponent interactions, while bulk models target properties such as phase formations, crystal structures, elastic constants, and yield strengths.
7 Conclusions: ML atomic interaction models help avoid time-consuming first-principles thermodynamics simulations and enable near-DFT-accuracy simulations at otherwise inaccessible scales.Their efficiency and accuracy support large-scale Monte Carlo and molecular-dynamics studies of HEA thermodynamics and mechanical properties.
7 Conclusions: Bulk-property ML models use experimental or theoretical datasets and require feature selection, model regularization, and out-of-sample validation.Neural networks, support-vector machines, Gaussian processes, and classification-and-regression trees are among the methods employed.
7 Conclusions: ML models have demonstrated excellent performance for predicting HEA phase formations and mechanical properties, and have guided designs whose exceptional mechanical properties were experimentally validated.These applications exploit the vast compositional space and complex chemical interactions that limit empirical rules.
7 Conclusions: Future research directions include uncertainty quantification, end-to-end descriptors, and ML-assisted inverse materials design.The review presents these directions as important opportunities for further research in ML-assisted HEA studies.