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

1.1 High-entropy alloys

• HEAs mix multiple principal elements into single-phase random solid solutions, opening a vast compositional space; some overcome the strength–ductility tradeoff.Their configurations also span short- and long-range order, which affects deformation-related physical quantities; tuning order can improve mechanical properties.Source: §1.1 · pp. 3–4
• Configurational entropy was proposed as a stabilizing factor for HEAs, but its role is not always as important as originally assumed.HEAs also offer a platform for studying how chemical order and disorder relate to deformation mechanisms.Source: §1.1 · pp. 3–4

1.2 Challenges in modeling high-entropy alloys

• HEA modeling faces a large space of chemical interactions that can make traditional interaction models prone to overfitting, while DFT and trial-and-error screening are costly.Even first-four-shell pair interactions yield 40 types in a five-element system; DFT studies may require supercells exceeding 1,000 atoms and many configuration calculations.Source: §1.2 · pp. 4–5

1.3 Merits of machine learning for high-entropy alloys

• ML models can simulate systems with millions of atoms near DFT accuracy and learn features or latent variables for predicting HEA properties [62].The review notes applications in atomistic simulations [68] [69], physical-property prediction [71], and materials design.Source: §1.3 · p. 6

1.4 Uniqueness and structure of this review

• The authors identify a lack of comprehensive reviews on ML for HEAs despite existing broader reviews of ML for materials science.This review focuses on the progress, challenges, and opportunities of applying ML to HEAs.Source: §1.4 · p. 7
• The review comprehensively surveys ML methods and applications for HEAs, explaining basic models and illustrating their advantages.Source: §1.4 · p. 8
• It examines HEA theoretical-modeling challenges and how ML can address them from microscopic and macroscopic perspectives.Source: §1.4 · p. 8
• The review compares the strengths and shortcomings of ML methods for HEAs and discusses remaining challenges and future opportunities.Source: §1.4 · p. 8
• It aims to guide researchers by surveying ML applications in multicomponent alloys and elucidating unresolved challenges.Source: §1.4 · p. 8

• ML is most effective for problems with abundant high-dimensional data and well-defined objectives, so HEA applications begin by confirming the problem can be converted to an ML task.Source: loys? · pp. 8–9
• Collected data are generally divided into training, validation, and test sets for model training, model selection, and performance evaluation, respectively.Source: loys? · p. 9, Figure 2 · p. 9
• A well-trained model should capture underlying physics, which can be explored by inspecting model parameters or applying dimension-reduction techniques.Source: loys? · p. 9, Figure 2 · p. 9
• A typical HEA machine-learning workflow identifies an application, selects an appropriate algorithm, collects data, trains and selects a model, and analyzes the results.Examples include accelerating simulations, predicting properties, and extracting hidden physics; algorithms include NN, SVM, GP, CART, and GAN, while model selection may use grid search or Bayesian optimization.Source: Figure 2 · p. 9, loys? · p. 9

2.1 Application identification

• ML surrogate applications divide into atomic interaction models (AIMs) for atomistic simulations and bulk property models (BPMs) for macro-scale physical-property predictions.They address materials-science problems at different time and length scales.Source: §2.1 · p. 10
• AIMs use atomic arrangements as inputs and offer high transferability, whereas BPMs directly predict target properties without expensive simulations.A well-trained AIM can evaluate multiple quantities, including formation energies, elastic constants, stacking-fault energies, and phonon spectra [59].Source: §2.1 · p. 10
• AIMs require greater predictive accuracy than BPMs because a nonphysical force in one molecular-dynamics step can derail the subsequent trajectory.This motivates incorporating physical information such as translational, rotational, and permutational symmetry.Source: §2.1 · p. 10
• For some models, including decision trees and shallow neural networks, inspecting weight parameters identifies essential features; clustering groups many features for dimension reduction, helping establish relations between desired properties and a few labeled data.Source: §2.1 · pp. 10–11
• A variational-autoencoder-based order parameter describes phase transitions in HEAs.The VAE reduces chemical-configuration data to a low-dimensional latent space; the configurations were generated by Monte Carlo simulations informed by ML-derived interaction parameters.Source: §2.1 · p. 11

Neural network (NN)

• Neural networks stack layers of neurons that weight prior-layer inputs, add bias, and apply activations; SGD with backpropagation commonly updates parameters to minimize loss.Source: §2.2 · p. 12

Support vector machine (SVM)

• SVM separates groups with a maximum-margin hyperplane determined by closest points; soft margins handle nonseparable data, while kernels extend boundaries to nonlinear cases.Source: §2.2 · pp. 12–13

Gaussian process (GP)

• GP estimates model-parameter probability distributions, enabling uncertainty quantification, but becomes computationally expensive above 10,000 data points and may require subsampling for sparsity.Source: §2.2 · p. 13

Classification and regression tree (CART)

• CART decision trees are intuitive and interpretable, recursively splitting feature-based subsets using criteria based on a randomness measure.Source: §2.2 · pp. 13–14
• Ensembles strengthen weak single-tree predictors through bootstrap bagging with random-feature splits (random forest) or sequential boosting.Source: §2.2 · p. 14

2.3.1 Data for atomistic simulations

• Atomistic-surrogate training data should be homogeneous, sufficiently large, and representative: consistent DFT methods and parameters isolate configuration effects, while sampling configuration space supports representativity.Relevant DFT parameters include the exchange-correlation functional, plane-wave energy cutoff, and k-point mesh.Source: §2.3.1 · p. 14
• Linear-scaling DFT reduces complexity from O(N^3) to O(N), aiding >100-atom HEA supercells; Liu et al. [104] [105] [68] [69] used LSMS local interaction zones while retaining long-range electrostatics.LSMS restricts electron scattering to a local interaction zone, but evaluates long-range electrostatic interactions throughout.Source: §2.3.1 · pp. 14–15

2.3.2 Data for physical-property models

• Bulk-property training draws on experimental or simulation data; Pauling File and MEAD are experimental resources, and Pei et al. reported good ML performance using elemental data.Source: §2.3.2 · p. 15
• HEA datasets included 122 refractory HEAs, 370 HEAs, and a 1,545-record update; Gao et al. supplied 1,252 multicomponent data used by Pei et al. for phase prediction.Source: §2.3.2 · p. 16

2.3.3 Data augmentation

• DFT datasets are typically limited to tens of thousands despite enormous alloy configuration spaces, so active learning iteratively adds simulation points selected by energy-range, trajectory, or prediction-error criteria.These strategies target predicted energies outside the original range, frequently visited trajectory regions, or errors above a threshold.Source: §2.3.3 · pp. 16–17
• GAN augmentation by Lee et al. significantly improved phase-prediction performance, with test accuracy enhanced from 84.75%.The supplied passage gives the starting accuracy but not the enhanced value.Source: §2.3.3 · p. 17

2.4 Model selection

• Because feature relevance is unknown beforehand, a broad feature pool can avoid omitting essential descriptors but also creates redundancy and dimensionality problems that feature selection can mitigate.The stated purpose is to reduce model complexity and overfitting risk.Source: §2.4 · p. 17
• Feature selection can rank descriptors by independence and target correlation, or use LASSO regularization to penalize non-zero coefficients and produce a sparse model.Correlation coefficients, mutual information, and similarity scores are examples of descriptor measures.Source: §2.4 · pp. 17–18
• A more computationally intensive feature-selection approach trains models on different feature sets and chooses the best set using performance on a hold-out set.Source: §2.4 · p. 18
• Hyperparameters can be selected using information criteria such as BIC and AIC, or hold-out samples such as cross-validation to measure performance and tune them.Source: §2.4 · p. 18
• Grid and random search are common simple tuning methods; more involved options include genetic algorithms [134, 71], evolution strategies, gradient-based optimization, and Bayesian optimization, including Gaussian process and TPE methods.Source: §2.4 · p. 18

• Atomic-interaction models are trained against DFT total energies; effective Hamiltonians use only total energy, whereas interatomic potentials also require atomic forces.Source: §2.5 · p. 18
• The energy-and-force MSE loss is minimized using methods including linear regression, Newton’s method, SGD, and Adam [139].The force weight can be adjusted, for example according to the number of atoms in a configuration’s supercell.Source: §2.5 · p. 19
• DFT local-energy fitting could expand training examples from Nconfig to approximately Nconfig × N_k, attractive for large-supercell HEAs and feasible with LSMS Voronoi integration.Source: §2.5 · p. 19
• Neighboring local energies are highly correlated: charge transfer can raise one atom’s energy and lower another’s, while the total-energy change is much smaller.Source: §2.5 · p. 19
• Thus, small relative errors in atomic energies can still produce large relative errors in averaged total energy, whose standard deviation is smaller than that of local energies.Source: §2.5 · p. 19

• Hold-out samples absent from training are used to evaluate models, and testing samples should be representative and uncorrelated to avoid spuriously high test scores.Source: §2.6 · pp. 19–20
• Classification performance is commonly measured with recall, precision, AUC, or ROC curves, while regression uses RMSE and the R2 score.Source: §2.6 · p. 20
• ML predictions can yield physics insights, with interpretation generally easier for some models than others.Source: §2.6 · p. 20
• Decision-tree feature importance can be assessed through Gini-index reduction, while permutation importance measures performance loss after randomly shuffling a feature’s values.Source: §2.6 · p. 20
• Deep neural networks are difficult to interpret because of their nonlinearity and high dimensionality; for simple networks, Taylor expansion can approximate input sensitivity by linearizing the model.Source: §2.6 · p. 20
• t-SNE [142] maps high-dimensional data to two- or three-dimensional space while preserving similar neighbors, enabling latent-variable visualization for classification.Source: §2.6 · p. 20
• In a t-SNE visualization, well-separated data with different labels indicate that the model can distinguish them, whereas poor separation indicates difficulty.Source: §2.6 · p. 20

3 Atomic interaction models

• Models trained on small systems can transfer to larger ones by mapping each atom’s local environment to its energy contribution; forces, when needed, are fitted simultaneously with local energies.The atomic energy represents an individual atom’s contribution, while a long-range electrostatic term depends on chemical concentration rather than atomic arrangement.Source: §3 · p. 21

3.1 Symmetry-preserving descriptors

• Positions cannot be directly used as features when squeezed into one-dimensional vectors; neighboring-atom radial distances rij used as input ignore angular distributions, a drawback mitigated by adding angle θijk between the two lines.Source: §3.1 · p. 21

3.2 Machine-learning potentials

• ML potentials use flexible rather than fixed forms and data-driven training, validation, and tests on first-principles datasets; high-quality potentials can approach quantum-mechanical accuracy such as DFT and describe systems too complex for traditional potentials.Source: §3.2 · p. 22

3.2.1 High-dimensional neural network potential (HDNNP)

• HDNNP uses neural networks to map atomic environments to atomic energies, sums those energies for the system total, and trains against DFT results by iterative loss minimization.Separate networks represent each species, with the same network used for atoms of a given species and energy differences arising from their environments.Source: §3.2.1 · pp. 22–23

3.2.2 Gaussian approximation potential (GAP)

• GAP uses Gaussian process regression to compare atomic environments, while sparse representation reduces its cost from O(N^3) to O(N^2M) when N ≫ M [147, 144, 155, 156, 158].Its SOAP kernel is derived from overlap of neighboring atomic densities and is used in the Gaussian-process regression.Source: §3.2.2 · pp. 23–24

3.2.3 Spectral neighbor analysis potential (SNAP)

• SNAP represents atomic neighbor density with spherical harmonics and constructs rotation-invariant power-spectrum or bispectrum descriptors for regression models, including linear SNAP and qSNAP.The density is mapped within the cutoff radius to a sphere in four-dimensional space.Source: §3.2.3 · pp. 24–25

3.2.4 Moment tensor potential (MTP)

• MTP constructs invariant descriptors by contracting moment tensors and expresses total energy using coefficients determined from a dataset.The supplied passage introduces the coefficients as quantities to be fitted from the dataset.Source: §3.2.4 · pp. 25–26
• MTP radial functions select atomic shells, while position-vector tensors encode angular information; contracted weighted moments form basis functions whose parameters are fitted by minimizing training loss.The moment level is set by μ and ν, with all tensor moments included up to the chosen level cutoff.Source: §3.2.4 · p. 26

3.3 Effective rigid-lattice Hamiltonian

• In a rigid-lattice model, atomic positions are not relaxed; lattice total energy can be expanded in cluster interactions, whose effective cluster interactions are fitted to DFT energies of selected configurations.Source: §3.3 · pp. 26–27
• For HEAs, conventional cluster expansion faces many interactions and costly accurate parameter extraction, making overfitting likely and structure-inversion estimates risky.A data-driven approach with regularization and model selection is proposed as a route to more robust models.Source: §3.3 · p. 27
• Regularized regression, including Bayesian ridge and LASSO variants, and model selection using BIC or cross-validation can help balance fit accuracy and robustness.Group LASSO is proposed for multicomponent systems.Source: §3.3 · pp. 27–28
• Because low-order linear expansions discard higher-order interactions that scale as N m in m-component HEAs, kernel ridge regression, quadratic regression, or neural networks can model nonlinear effects.Natarajan and Van der Ven developed a nonlinear formalism for calculating atomic energy using local correlation functions.Source: §3.3 · p. 28
• With six local features, the neural network test error was 0.006 eV/atom versus 0.01 eV/atom for linear regression, and it consistently performed better, especially with few features.The neural network had three layers with 4, 4, and 2 nodes, and was trained on 1000 configurations and tested on 1346.Source: §3.3 · pp. 28–29

4.1 Monte-Carlo simulations

• For MoNbTaW, MoNbTaVW, and MoNbTaTiW, data-driven effective pair-interaction Hamiltonians had error bars below 0.1 mRy and were validated against 1000-atom MC configurations, supporting active learning with MC configurations [69].The validation used total energies computed with the linear-scaling LSMS method.Source: §4.1 · p. 32
• The MC calculations identified two order-disorder transition temperatures for each of the three refractory HEAs and found lower transition temperatures for MoNbTaW than for MoNbTaVW [69].The transitions were identified from specific heats and short-range-order parameters.Source: §4.1 · p. 32, Figure 5 · p. 33
• Between 1000 and 2000 K, MoNbTaW is expected to be more solid-solution-rich and more ductile than MoNbTaTiW, with the ductility expectation confirmed experimentally [69].MoNbTaTiW is expected to contain a large fraction of second-phase precipitates; the passage also compares experimental peak strains for MoNbTaW and MoNbTaVW.Source: §4.1 · p. 32, Figure 5 · p. 33
• For MoNbTaW, lattice relaxation is important below 500 K; the lattice-distortion effect ignored in the rigid-lattice model was included using DFT-relaxed structures in studying order-disorder transitions.Source: §4.1 · p. 32

4.2 Molecular-dynamics simulations

• MTP-informed MD separated static, dynamic, thermal-expansion, and chemical short-range-order contributions to local lattice distortion in CoFeNi and assessed their effects on elastic properties.The simulations used configurations from a sample melted at 2300 K, quenched to 300 K, and equilibrated for 5 ns.Source: §4.2 · pp. 33–34
• MTP-informed MD showed that structural phase changes can profoundly affect the elastic properties of TiZrHfTa as Ta concentration is varied.Source: §4.2 · p. 34
• For the VZrNbHfTa melt, HDNNP and ab initio MD gave good agreement in partial radial distribution functions, while vanadium’s dislike of other species lowered the likelihood of a single-phase solid solution.Source: §4.2 · p. 34

4.3 Hybrid MC/MD simulations

• Generalized stacking fault energy (GSFE) characterizes plastic deformation by determining dislocation core structure and the minimum stress for dislocation motion, but chemical disorder requires evaluating many configurations.Efficient ML potentials are therefore desirable to accelerate GSFE calculations for multicomponent alloys.Source: §4.3 · p. 34
• In MoNbTaW, SNAP-based MC/MD found GSFEs closer to W and Mo than Ta and Nb, with Peierls stresses of 1620 ± 637 MPa for screw and 320 ± 113 MPa for edge dislocations [65].The calculations used a 36,000-atom supercell, beyond conventional DFT capability.Source: §4.3 · pp. 34–35
• For Mo-Nb-Ta alloys, MTP predicted unstable stacking-fault energies with a maximum relative error of 6-7% versus DFT, while active learning reduced training data from about 10,000 to 50-100 configurations.The authors proposed this data efficiency as attractive for high-throughput screening of multicomponent alloys.Source: §4.3 · pp. 35–36
• For Al-Cu precipitate GSFEs, HDNNP results agreed with experiment better than the angular-dependent EAM potential.The comparison covered θ and θ′′ precipitates and also included DFT.Source: §4.3 · p. 36
• In MoNbTaWV radiation-damage simulations, a tabulated Gaussian approximation potential found Mo-Ta and V-W ordering in the bulk and vanadium segregation to compressed, interstitial-rich damaged regions.The Mo-Ta ordering was consistent with other studies [69].Source: §4.3 · p. 36
• For BCC iron, GAP matched DFT GSFEs and outperformed EAM; its Peierls stress was about 2 GPa, consistent with DFT, while SNAP gave reasonable DFT agreement for Mo, Ta, Nb, and W [178][179][180].Source: §4.3 · p. 36

5.1 Prediction of solid solutions and multi-phases

• ML phase prediction spans solid-solution, intermetallic, amorphous, and mixed phases, while crystal-structure models distinguish structures across compositional spaces of up to 16 elements.Tancret et al. combined previously proposed criteria to guide solid-solution HEA design.Source: §5 · p. 37, §5.1 · p. 40
• The genetic-algorithm-selected model achieved 88.7% accuracy for solid-solution versus non-solid-solution phases and 91.3% for BCC, FCC, and dual-phase HEAs.The study screened 45 models and selected both the model and material descriptors from a larger pool.Source: §5.1 · p. 39
• The ANN achieved 74.3% accuracy for predicting solid-solution, intermetallic, and mixed phases, compared with KNN’s 68.6%.The comparison used an experimental dataset of 401 HEAs; the supplied passage also reports 64.3% but does not identify its model.Source: §5.1 · p. 39
• The optimized neural network reached 84.75% accuracy, outperforming SVM, decision tree, and XGBoost models.Conditional GAN data augmentation improved the neural-network model’s performance.Source: §5.1 · p. 40
• Gradient-boosting decision trees predicted phases and Young’s modulus with generally good agreement across dozens of alloys in the Mo-Ta-Ti-W-Zr family.Random forests were also evaluated for predicting solid-solution formation.Source: §5.1 · p. 40

5.2 Machine-learning informed new rules for phase predictions

• ANN models classified elemental roles in Al-Co-Cr-Fe-Ni and predicted near-eutectic compositions, supporting a new rule of thumb for HEA design.The database combined literature results and CALPHAD calculations for 321 alloys.Source: §5.2 · pp. 40–41
• A Gaussian Process Classification example uses prediction distributions to mitigate overfitting and handle small datasets while constructing a thermodynamics-based rule for solid-solution formation.The model’s predictive variance is smaller with more data and larger with less data.Source: §5.2 · p. 41
• The example trained on 1,252 alloys with known BCC, FCC, HCP, or multiphase status, using 85 elemental properties averaged over alloy compositions.Single-phase alloys were labeled 1 and multiphase alloys 0.Source: §5.2 · p. 41
• GPC achieved AUC 0.97 (0.96–0.97 across crystal structures) and 93% validation accuracy, with probability >0.5 identifying solid solutions.Half of the 1,252 alloys were used for training and the remainder for validation.Source: §5.2 · p. 41, Figure 8 · p. 42
• Compared with the ∆H empirical rule, ML uses high-dimensional features; its thermodynamics-informed γ ≥1 plus lattice-distortion δ rule reached 75% accuracy and supports transparent high-throughput screening.The empirical ∆H window is inaccurate because many multiphase alloys also fall within it.Source: Figure 8 · p. 42, §5.2 · p. 43

5.3 Machine-learning based order parameters

• A VAE’s 12D latent embedding separates phases, enabling a Manhattan-distance scalar order parameter that tracks B2-to-A2 ordering as temperature rises.The demonstrated sequence is strong B2 order, partial B2 order, then A2 (BCC).Source: §5.3 · p. 44, Figure 9 · p. 46
• Its susceptibility correctly signals HEA transition temperatures and captures classical ordering trends, aligning with LRO at high temperature and SRO at low temperature without simply reproducing either.The new order parameter has features distinct from conventional order parameters.Source: §5.3 · p. 45, Figure 9 · p. 46

5.4 Mechanical-property prediction

• In a binary Mg-alloy example, descriptors tied to non-basal-dislocation nucleation and mobility explain observations and predict enhanced ductility, with extension to HEAs proposed.The passage notes limited experimental training data and few representative HEA examples, particularly for ductility.Source: §5.4 · p. 47
• Tensile tests confirmed enhanced ductility predicted by DFT for Mg alloys, while ML predictions agreed with two descriptors.Figure 10(c) compares ML with the YSI descriptor (Mechanism I) and −δX (Mechanism II); pink regions indicate agreement.Source: §5.4 · p. 47, Figure 10 · p. 48
• When descriptors disagreed, ML could help identify the more promising mechanism, and ML results could be used to rediscover a descriptor for Mechanism I [50].A mechanism whose descriptors agree better with ML is more likely to be active during deformation.Source: §5.4 · pp. 47, 49
• CCA and a genetic algorithm identified promising hard HEAs experimentally confirmed to have exceptional hardness, while ML-guided alloys achieved hardness 10% above the best training alloy.Wen et al. compared several regression and classification approaches before synthesizing the higher-hardness alloys.Source: §5.4 · p. 49
• Using actual processing conditions and CALPHAD information, ML models reproduced existing experimental yield stresses for high-temperature multicomponent steels; models also predicted MoNbTaTiW and HfMoNbTaTiZr yield stresses at 800, 1200, and 1500 °C.Source: §5.4 · p. 49
• For a designed HEA, experiments validated a yield strength of 1.31 GPa and tensile elongation of 15%.An artificial neural network predicted γ′ phase volume and yield stress from composition, microstructure, and properties, guiding exploration of compositional space.Source: §5.4 · pp. 49–50
• Including lattice distortion in DFT calculations brought elastic constants for FCC AlCoCrFeNi closer to experiments, and an ML model trained on inorganic structures accurately predicted HEA elastic properties.Source: §5.4 · p. 50

6.1 Challenges

• ML predictions require confidence assessment, and objective, extensive benchmarks could help establish heuristic rules for model performance and applicability.The review notes that trusting predictions without knowing when they are accurate can be risky, as illustrated by adversarial attacks.Source: §6.1 · p. 50
• The high-dimensional feature space motivates end-to-end learning of physics-relevant descriptors, but flexible models require more data and progress in both modeling and sampling.Graph neural networks represent neighboring atoms and bonds as graph nodes and edges; Ref. reports that GNN methods require one or two magnitudes larger datasets for similar accuracy to descriptor methods.Source: §6.1 · pp. 50–51
• High-quality HEA datasets remain difficult to build because measurements are costly, conditions vary across groups, and the vast design space leaves datasets small.Combining data requires careful calibration, and existing dataset-improvement methods do not fully solve the small-data problem.Source: §6.1 · p. 51
• Researchers are encouraged to publish failed trials alongside successful results and report measurement conditions so property data can be cleaned and calibrated consistently.Measured properties depend on synthesis, heat treatment, and measurement methods; omitting these details can compromise dataset quality and model performance.Source: §6.1 · pp. 51–52

6.2 Uncertainty quantification

• Uncertainty quantification is important for assessing prediction reliability in HEAs, where datasets are typically small relative to the huge design spaces.Confidence intervals can also support active learning by directing data collection toward regions of high uncertainty.Source: §6.2 · pp. 52–53
• Uncertainty quantification methods include Dempster-Shafer evidence theory, fuzzy sets, interval methods, information theory, Bayesian approaches, and Monte Carlo methods.Monte Carlo methods are straightforward to implement but typically computationally intensive; modern variants address this shortcoming.Source: §6.2 · p. 53

6.3 End-to-end (automatically learned) descriptors

• Graph convolutional neural networks preserve atom and bond concepts in graph-structured data and demonstrate excellent interpretability.CNN-based models can automatically learn important features, but their complexity generally requires more training data.Source: §6.3 · p. 54
• End-to-end descriptors are rarely used for HEA atomistic simulations because their overfitting risk demands more extensive training data, despite their greater representational capacity.The outputs of a graph convolutional neural network’s final convolutional layer can be interpreted as local atomic energies.Source: §6.3 · p. 54

6.4 Machine-learning guided inverse materials design

• Multiple alloy and processing combinations can yield the same properties, making inverse materials design challenging.The passage attributes this difficulty to the reduction from many design factors to targeted properties and the resulting asymmetry of information flow and lack of constraints.Source: §6.4 · p. 55
• A neural network separates steels by composition and mechanical properties, including room-temperature yield stress, and supports an inverse-design example.The example uses steels with about 20 components because typical HEA examples are lacking; the authors state that the method is applicable to HEAs.Source: §6.4 · pp. 55–56, Figure 12 · p. 57
• The inverse designer reverses a trained regression model’s information flow to find an optimal microstructure and map it to alloy components.The approach can be applied to other material systems given sufficient microstructure images.Source: §6.4 · p. 56
• CNNs identified L1_2-ordered nanostructures in FCC Al-Li-Mg APT data, while ResNet50 and Xception each classified EBSD crystal structures with >90% accuracy.The APT task is difficult for researchers because it involves millions of data points.Source: §6.4 · p. 56
• Scientific-image analysis with ML remains in its infancy, but with method improvement and more experimental data it may predict mechanical properties directly from microstructure images.A fully convolutional network inferred atomic positions from STEM image pixel intensities with sufficiently high confidence.Source: §6.4 · pp. 56, 58

• DFT-based atomic-interaction models represent complex multicomponent interactions and help avoid time-consuming first-principles thermodynamics simulations.Source: §7 · p. 58
• Symmetry-preserving ML models, including HDNNP, GAP, SNAP, MTP, LRP, and Bayesian CE, enable near-DFT-accuracy simulations at scales inaccessible to conventional methods [62].Source: §7 · p. 58
• ML atomic-interaction models have been used in Monte Carlo or molecular dynamics simulations to investigate HEA thermodynamic and mechanical properties.Source: §7 · p. 58
• Bulk-property models use experimental or calculated datasets, with feature selection, model choice, regularization, and out-of-sample validation as key procedures.Source: §7 · pp. 58–59
• ML models predict HEA phase formations and mechanical properties, and have guided alloy designs with exceptional mechanical properties validated experimentally.Source: §7 · p. 59
• Promising research directions include uncertainty quantification, end-to-end descriptors, and ML-assisted inverse materials design.Source: §7 · p. 59