Acta Metallurgica Sinica (English Letters) ›› 2025, Vol. 38 ›› Issue (8): 1261-1274.DOI: 10.1007/s40195-025-01876-4
Jiayu Wang1, Ke Liu2, Zhao Lei1, Xing Li2, Li Liu1(
), Sujun Wu2()
Received:2024-12-02
Revised:2025-01-24
Accepted:2025-02-13
Online:2025-08-10
Published:2025-05-24
Contact:
Li Liu, Sujun Wu
Jiayu Wang, Ke Liu, Zhao Lei, Xing Li, Li Liu, Sujun Wu. Machine-Learning-Assisted Phase Prediction in High-Entropy Alloys Using Two-Step Feature Selection Strategy[J]. Acta Metallurgica Sinica (English Letters), 2025, 38(8): 1261-1274.
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| Features | Formula |
|---|---|
| Average atomic radius | $a= \sum_{i=1}^{n}{c}_{i}{r}_{i}$ |
| Atomic size difference | $\delta = \sqrt{\sum_{i=1}^{n}{c}_{i}{(1-\frac{{r}_{i}}{a})}^{2}}$ |
| Melting point | ${T}_{\text{m}}= \sum_{i=1}^{n}{c}_{i}{T}_{i}$ |
| Standard deviation of melting point | ${\sigma }_{T}= \sqrt{\sum_{i=1}^{n}{c}_{i}{(1-\frac{{T}_{i}}{{T}_{\text{m}}})}^{2}}$ |
| Mixing enthalpy | $\Delta {H}_{\text{mix}}=\sum_{i=1, i\ne j}^{n}4{H}_{ij}{c}_{i}{c}_{j}$ |
| Standard deviation of mixing enthalpy | ${\sigma }_{\Delta H}= \sqrt{\sum_{i=1, i\ne j}^{n}{c}_{i}{c}_{j}{({H}_{ij}-\Delta {H}_{\text{mix}})}^{2}}$ |
| Mixing entropy | $\Delta {S}_{\text{mix}}= -R\sum_{i=1}^{n}{c}_{i}\text{ln}{c}_{i}$ |
| Electronegativity | $\chi = \sum_{i=1}^{n}{c}_{i}{\chi }_{i}$ |
| Electronegativity difference | $\Delta \chi = \sqrt{\sum_{i=1}^{n}{c}_{i}{({\chi }_{i}-\chi )}^{2}}$ |
| Valence electron concentration | $\text{VEC}= \sum_{i=1}^{n}{c}_{i}{\text{VEC}}_{i}$ |
| Standard deviation of valence electron concentration | ${\sigma }_{\text{VEC}}= \sqrt{\sum_{I=1}^{n}{c}_{i}{({\text{VEC}}_{i}-\text{VEC})}^{2}}$ |
Table 1 Eleven features selected for building machine learning models
| Features | Formula |
|---|---|
| Average atomic radius | $a= \sum_{i=1}^{n}{c}_{i}{r}_{i}$ |
| Atomic size difference | $\delta = \sqrt{\sum_{i=1}^{n}{c}_{i}{(1-\frac{{r}_{i}}{a})}^{2}}$ |
| Melting point | ${T}_{\text{m}}= \sum_{i=1}^{n}{c}_{i}{T}_{i}$ |
| Standard deviation of melting point | ${\sigma }_{T}= \sqrt{\sum_{i=1}^{n}{c}_{i}{(1-\frac{{T}_{i}}{{T}_{\text{m}}})}^{2}}$ |
| Mixing enthalpy | $\Delta {H}_{\text{mix}}=\sum_{i=1, i\ne j}^{n}4{H}_{ij}{c}_{i}{c}_{j}$ |
| Standard deviation of mixing enthalpy | ${\sigma }_{\Delta H}= \sqrt{\sum_{i=1, i\ne j}^{n}{c}_{i}{c}_{j}{({H}_{ij}-\Delta {H}_{\text{mix}})}^{2}}$ |
| Mixing entropy | $\Delta {S}_{\text{mix}}= -R\sum_{i=1}^{n}{c}_{i}\text{ln}{c}_{i}$ |
| Electronegativity | $\chi = \sum_{i=1}^{n}{c}_{i}{\chi }_{i}$ |
| Electronegativity difference | $\Delta \chi = \sqrt{\sum_{i=1}^{n}{c}_{i}{({\chi }_{i}-\chi )}^{2}}$ |
| Valence electron concentration | $\text{VEC}= \sum_{i=1}^{n}{c}_{i}{\text{VEC}}_{i}$ |
| Standard deviation of valence electron concentration | ${\sigma }_{\text{VEC}}= \sqrt{\sum_{I=1}^{n}{c}_{i}{({\text{VEC}}_{i}-\text{VEC})}^{2}}$ |
Fig. 1 Data visualization. a Pair plot showing the relationship between phase structure and all design parameters in the 296 datasets. b Hexagonal binning for $\text{VEC}$ parameter used for phase structure prediction
Fig. 2 First step feature selection. a Pearson correlation coefficient with eleven features. b Cross-validation accuracy with $\text{VEC}$ or $a$ feature
Fig. 3 Second step feature selection. a Importance ranking of ten features using RF model. The cross-validation accuracy of b MLP, c SVM, d DT, e RF, and f KNN model with various features
| Model | Selected features |
|---|---|
| MLP | $\text{VEC}$ |
| SVM | $\text{VEC}$, ${T}_{\text{m}}$, ${\sigma }_{\text{VEC}}$, ${\sigma }_{T}$, $\chi $, $\Delta \chi $, $\delta $, $\Delta {H}_{\text{mix}}$ |
| DT | $\text{VEC}$, ${T}_{\text{m}}$ |
| RF | $\text{VEC}$, ${T}_{\text{m}}$, ${\sigma }_{\text{VEC}}$ |
| KNN | $\text{VEC}$, ${T}_{\text{m}}$ |
Table 2 Finally selected features for building machine learning models
| Model | Selected features |
|---|---|
| MLP | $\text{VEC}$ |
| SVM | $\text{VEC}$, ${T}_{\text{m}}$, ${\sigma }_{\text{VEC}}$, ${\sigma }_{T}$, $\chi $, $\Delta \chi $, $\delta $, $\Delta {H}_{\text{mix}}$ |
| DT | $\text{VEC}$, ${T}_{\text{m}}$ |
| RF | $\text{VEC}$, ${T}_{\text{m}}$, ${\sigma }_{\text{VEC}}$ |
| KNN | $\text{VEC}$, ${T}_{\text{m}}$ |
Fig. 4 Cross-validation accuracy of a MLP, b SVM, c DT, d RF, and e KNN model containing a subset of initial 11 features
Fig. 5 Hyperparameter optimization of a MLP, b SVM, c RF, d DT, and e KNN model. f The cross-validation accuracy after optimization
| Model | Hyperparameters |
|---|---|
| MLP | First layer number = 5, Second layer number = 9 |
| SVM | C = 13, γ = 1 |
| DT | Max depth = 4 |
| RF | Max depth = 9, Estimators = 47 |
| KNN | Neighbors = 1 |
Table 3 Hyperparameters of five machine learning models after optimization
| Model | Hyperparameters |
|---|---|
| MLP | First layer number = 5, Second layer number = 9 |
| SVM | C = 13, γ = 1 |
| DT | Max depth = 4 |
| RF | Max depth = 9, Estimators = 47 |
| KNN | Neighbors = 1 |
Fig. 6 a Prediction accuracy of traditional $\text{VEC}$ criterion and machine learning methods in 296 datasets. Confusion matrixes of b MLP, c SVM, d DT, e RF, and f KNN model for testing dataset
Fig. 7 Parallel coordinate plot shows the distribution of final selected features for BCC, FCC, and BCC + FCC phase structure
| Composition | VEC | MLP | SVM | DT | RF | KNN | Experiment |
|---|---|---|---|---|---|---|---|
| Al1CoCu6Ni6Fe6 | FCC | FCC | FCC | FCC | FCC | FCC | FCC |
| Al3CoCu6Ni6Fe6 | FCC | FCC | FCC | FCC | FCC | FCC | FCC |
| Al6CoCu6Ni6Fe6 | FCC | FCC | BCC + FCC | FCC | BCC + FCC | BCC + FCC | BCC + FCC |
| CoCuFeNi | FCC | FCC | FCC | FCC | FCC | FCC | FCC |
| CoCuFeNi5 | FCC | FCC | FCC + BCC | FCC | FCC + BCC | FCC | FCC + BCC |
| CoCuFeNi10 | FCC | FCC | FCC | FCC | FCC + BCC | FCC | FCC + BCC |
| CoCuFeNi15 | FCC | FCC | FCC | FCC | FCC | FCC | FCC |
| CoCuFeNi20 | FCC | FCC | FCC | FCC | FCC | FCC | FCC |
| Ta0.45Nb0.35Zr0.10Ti0.10 | BCC | BCC | BCC | BCC | BCC | BCC | BCC |
| Ta0.60Nb0.15Zr0.05Ti0.20 | BCC | BCC | BCC + FCC | BCC | BCC | BCC | BCC |
| Ta0.05Nb0.70Zr0.15Ti0.10 | BCC | BCC | BCC | BCC | BCC | BCC | BCC |
| Ta0.40Nb0.30Zr0.05Ti0.25 | BCC | BCC | BCC | BCC | BCC | BCC | BCC |
| Ta0.05Nb0.65Zr0.05Ti0.25 | BCC | BCC | BCC + FCC | BCC | BCC | BCC | BCC |
Table 4 Experimental and prediction results of the phase structure of AlxCoCu6Ni6Fe6 (x = 1, 3, 6) HEAs with various machine learning models and $\text{VEC}$ criterion
| Composition | VEC | MLP | SVM | DT | RF | KNN | Experiment |
|---|---|---|---|---|---|---|---|
| Al1CoCu6Ni6Fe6 | FCC | FCC | FCC | FCC | FCC | FCC | FCC |
| Al3CoCu6Ni6Fe6 | FCC | FCC | FCC | FCC | FCC | FCC | FCC |
| Al6CoCu6Ni6Fe6 | FCC | FCC | BCC + FCC | FCC | BCC + FCC | BCC + FCC | BCC + FCC |
| CoCuFeNi | FCC | FCC | FCC | FCC | FCC | FCC | FCC |
| CoCuFeNi5 | FCC | FCC | FCC + BCC | FCC | FCC + BCC | FCC | FCC + BCC |
| CoCuFeNi10 | FCC | FCC | FCC | FCC | FCC + BCC | FCC | FCC + BCC |
| CoCuFeNi15 | FCC | FCC | FCC | FCC | FCC | FCC | FCC |
| CoCuFeNi20 | FCC | FCC | FCC | FCC | FCC | FCC | FCC |
| Ta0.45Nb0.35Zr0.10Ti0.10 | BCC | BCC | BCC | BCC | BCC | BCC | BCC |
| Ta0.60Nb0.15Zr0.05Ti0.20 | BCC | BCC | BCC + FCC | BCC | BCC | BCC | BCC |
| Ta0.05Nb0.70Zr0.15Ti0.10 | BCC | BCC | BCC | BCC | BCC | BCC | BCC |
| Ta0.40Nb0.30Zr0.05Ti0.25 | BCC | BCC | BCC | BCC | BCC | BCC | BCC |
| Ta0.05Nb0.65Zr0.05Ti0.25 | BCC | BCC | BCC + FCC | BCC | BCC | BCC | BCC |
Fig. 8 XRD profiles of AlxCoCu6Ni6Fe6 (x = 1, 3, 6) HEAs
Fig. 9 Backscattered electron micrographs of AlxCoCu6Ni6Fe6 HEAs: a x = 1, b x = 3, c x = 6; distribution of Cu element in AlxCoCu6Ni6Fe6 HEAs: d x = 1, e x = 3, f x = 6
| $\Delta {H}_{\text{mix}}^{ij}$ (kJ/mol) | Al | Co | Cu | Ni | Fe |
|---|---|---|---|---|---|
| Al | - | − 19 | − 1 | − 22 | − 11 |
| Co | − 19 | - | 6 | 0 | − 1 |
| Cu | − 1 | 6 | - | 4 | 13 |
| Ni | − 22 | 0 | 4 | - | − 2 |
| Fe | − 11 | − 1 | 13 | − 2 | - |
Table 5 Binary mixing enthalpies of atomic pairs in AlxCoCu6Ni6Fe6 (x = 1, 3, 6) HEAs
| $\Delta {H}_{\text{mix}}^{ij}$ (kJ/mol) | Al | Co | Cu | Ni | Fe |
|---|---|---|---|---|---|
| Al | - | − 19 | − 1 | − 22 | − 11 |
| Co | − 19 | - | 6 | 0 | − 1 |
| Cu | − 1 | 6 | - | 4 | 13 |
| Ni | − 22 | 0 | 4 | - | − 2 |
| Fe | − 11 | − 1 | 13 | − 2 | - |
Fig. 10 EBSD images of AlxCoCu6Ni6Fe6 (x = 1, 3, 6) HEAs. a, d Image quality (IQ) map and grain size statistics diagram of Al1CoCu6Ni6Fe6 HEA. b, e IQ image and grain size statistics diagram of Al3CoCu6Ni6Fe6 HEA. c IQ image and f grain size statistics diagram and g, h phase maps of Al6CoCu6Ni6Fe6 HEA
Fig. 11 a Engineering stress-strain curves of AlxCoCu6Ni6Fe6 (x = 1, 3, 6) HEAs. b Potentiodynamic polarization curves of AlxCoCu6Ni6Fe6 HEAs in 3.5% NaCl solution
Yield strength (MPa) | Tensile strength (MPa) | Elongation (%) | Hardness (HV) | |
|---|---|---|---|---|
| Al1CoCu6Ni6Fe6 | 517 ± 5 | 732 ± 17 | 20.4 ± 4.1 | 169 ± 3 |
| Al3CoCu6Ni6Fe6 | 654 ± 10 | 722 ± 12 | 1.8 ± 0.01 | 230 ± 5 |
| Al6CoCu6Ni6Fe6 | - | 509 ± 40 | 1.5 ± 0.3 | 513 ± 5 |
Table 6 Mechanical properties of AlxCoCu6Ni6Fe6 HEAs
Yield strength (MPa) | Tensile strength (MPa) | Elongation (%) | Hardness (HV) | |
|---|---|---|---|---|
| Al1CoCu6Ni6Fe6 | 517 ± 5 | 732 ± 17 | 20.4 ± 4.1 | 169 ± 3 |
| Al3CoCu6Ni6Fe6 | 654 ± 10 | 722 ± 12 | 1.8 ± 0.01 | 230 ± 5 |
| Al6CoCu6Ni6Fe6 | - | 509 ± 40 | 1.5 ± 0.3 | 513 ± 5 |
${I}_{\text{corr}}$ (μA/cm2) | ${E}_{\text{corr}}$ (mv) | |
|---|---|---|
| Al1CoCu6Ni6Fe6 | 0.206 ± 0.03 | -314 ± 19.6 |
| Al3CoCu6Ni6Fe6 | 0.370 ± 0.10 | -334 ± 54.6 |
| Al6CoCu6Ni6Fe6 | 0.649 ± 0.20 | -280 ± 34.1 |
Table 7 Electrochemical parameters of AlxCoCu6Ni6Fe6 HEAs in 3.5% NaCl solution
${I}_{\text{corr}}$ (μA/cm2) | ${E}_{\text{corr}}$ (mv) | |
|---|---|---|
| Al1CoCu6Ni6Fe6 | 0.206 ± 0.03 | -314 ± 19.6 |
| Al3CoCu6Ni6Fe6 | 0.370 ± 0.10 | -334 ± 54.6 |
| Al6CoCu6Ni6Fe6 | 0.649 ± 0.20 | -280 ± 34.1 |
Fig. 12 Tensile fracture morphology of AlxCoCu6Ni6Fe6 HEAs: a x = 1, b x = 3, c x = 6. Surface morphology of AlxCoCu6Ni6Fe6 HEAs after potentiodynamic polarization test in 3.5% NaCl solution: d x = 1, e x = 3, f x = 6
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