journal article·2026-01Open access

Interpretable machine learning for Co-free high-entropy alloy phase prediction: balancing the role of valence electron concentration

Wenyi Huo

Journal of Materials Science· Volume 61· Issue 4

Abstract

Abstract High-entropy alloys (HEAs) favor solid solution phases and exhibit superior properties in irradiation environments. However, the radiological risks of Co necessitate Co-free designs. In this work, different data augmentation techniques, i.e., synthetic minority oversampling technique (SMOTE) and Wasserstein generative adversarial network (WGAN) with gradient penalty, are evaluated for predicting body-centered cubic (BCC) phase stability in Co-free HEAs. Principal component analysis (PCA) reduces dimensionality. A simple multilayer perceptron trained on these augmented datasets achieved cross-validation accuracies of 84.39% (SMOTE) and 86.82% (WGAN). Shapley additive explanations (SHAPs) and PCA analyses show valence electron concentration’s dominance of SMOTE-generated data per Hume-Rothery rules. WGAN-generated data offer balanced thermodynamic contributions for enhanced generalizability. These results show that for predicting phases in materials with small datasets, simple algorithm supported by data augmentation techniques can be as effective as complex algorithms. This work can guide HEA design for niche-subset applications and broaden the utility of machine learning for small datasets.
Source: crossref · Quality: baseline

Research context

OpenAlex topics: High Entropy Alloys Studies · Machine Learning in Materials Science · Additive Manufacturing Materials and Processes

SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.

References (49)

Show source-provided references
1. Granberg F, Nordlund K, Ullah MW, Jin K, Lu C, Bei H, Wang LM, Djurabekova F, Weber WJ, Zhang Y (2016) Mechanism of radiation damage reduction in equiatomic multicomponent single phase alloys. Phys Rev Lett 116:13550410.1103/PhysRevLett.116.135504 ↗
2. George EP, Raabe D, Ritchie RO (2019) High-entropy alloys. Nat Rev Mater 4:515–53410.1038/s41578-019-0121-4 ↗
3. Huo W, Fang F, Liu X, Tan S, Xie Z, Jiang J (2019) Remarkable strain-rate sensitivity of nanotwinned CoCrFeNi alloys. Appl Phys Lett 114:10190410.1063/1.5088921 ↗
4. Gludovatz B, Hohenwarter A, Catoor D, Chang EH, George EP, Ritchie RO (2014) A fracture-resistant high-entropy alloy for cryogenic applications. Science 345:1153–115810.1126/science.1254581 ↗
5. Anber EA, Beaudry D, Brandenburg C, Marks LD, Thron AM, Vecchio K, Poerschke D, Hopkins P, Levi CG, Pollock TM (2024) Oxidation resistance of Al-containing refractory high-entropy alloys. Scr Mater 244:11599710.1016/j.scriptamat.2024.115997 ↗
6. Zhang F, Wu Y, Lou H, Zeng Z, Prakapenka VB, Greenberg E, Ren Y, Yan J, Okasinski JS, Liu X, Liu Y, Zeng Q, Lu ZP (2017) Polymorphism in a high-entropy alloy. Nat Commun 8:1568710.1038/ncomms15687 ↗
7. Lu C, Niu L, Chen N, Jin K, Yang T, Xiu P, Zhang Y, Gao F, Bei H, Shi S, He M-R, Robertson IM, Weber WJ, Wang L (2016) Enhancing radiation tolerance by controlling defect mobility and migration pathways in multicomponent single-phase alloys. Nat Commun 7:1356410.1038/ncomms13564 ↗
8. Su Z, Shi T, Yang J, Shen H, Li Z, Wang S, Ran G, Lu C (2022) The effect of interstitial carbon atoms on defect evolution in high entropy alloys. Acta Mater 233:11795510.1016/j.actamat.2022.117955 ↗
9. Wu W, Yang Q, Zhang W, Zhang Y, Yan D, Li Z (2025) Insights into irradiation-induced defect evolution and segregation in metastable high-entropy alloys: effects of high-density incoherent planar defects and temperature. Acta Mater 266:12099410.1016/j.actamat.2025.120994 ↗
10. Bloomfield M, Christofidou K, Jones N (2019) Effect of Co on the phase stability of CrMnFeCoxNi high entropy alloys. Intermetallics 114:10658210.1016/j.intermet.2019.106582 ↗
11. Li Y, Lu E, Kurpaska Ł, Fang F, Stasiak T, Kim HS, Weber WJ, Zhang Y, Huo W (2025) Mo-driven strengthening mechanisms in cobalt-free Cr20-xFe30Mn20Ni30Mox high-entropy alloys. Mater Sci Eng A 946:14915010.1016/j.msea.2025.149150 ↗
12. Li Y, Olejarz A, Kurpaska Ł, Lu E, Alava MJ, Kim HS, Huo W (2024) Designing cobalt-free face-centered cubic high-entropy alloys: a strategy using d-orbital energy level. Int J Refract Metal Hard Mater 124:10683410.1016/j.ijrmhm.2024.106834 ↗
13. Li Y, Kurpaska Ł, Lu E, Xie Z, Kim HS, Huo W (2024) Body-centered cubic phase stability in cobalt-free refractory high-entropy alloys. Res Phys 60:107688
14. Luo X, Li Y, Mäkinen T, Bonfanti S, Huo W, Alava MJ (2025) Data-driven body-centered cubic phase prediction in cobalt-free high-entropy alloys. Mater Today Commun 42:11246410.1016/j.mtcomm.2025.112464 ↗
15. Han D, Yang B, Xu W, Yang H, Han G, Wang X, Wang J (2024) Significantly ameliorating room-temperature brittleness of refractory high-entropy alloys via in situ heterogeneous structure. J Mater Sci Technol 193:1–1710.1016/j.jmst.2024.01.022 ↗
16. Guo Y, Peng J, Peng S, An F, Lu W, Li Z (2023) Improving oxidation resistance of TaMoZrTiAl refractory high entropy alloys via Nb and Si alloying. Corros Sci 223:11145510.1016/j.corsci.2023.111455 ↗
17. Liu Y, Wang J, Xiao B, Shu J (2022) Accelerated development of hard high-entropy alloys with data-driven high-throughput experiments. J Mater Inf 2:3
18. Zhu W, Huo W, Wang S, Wang X, Ren K, Tan S, Fang F, Xie Z, Jiang J (2022) Phase formation prediction of high-entropy alloys: a deep learning study. J Mater Res Technol 18:800–80910.1016/j.jmrt.2022.01.172 ↗
19. Liu X, Zhang J, Pei Z (2023) Machine learning for high-entropy alloys: progress, challenges and opportunities. Prog Mater Sci 131:10101810.1016/j.pmatsci.2022.101018 ↗
20. Rao Z, Tung PY, Xie R, Wei Y, Zhang H, Ferrari A, Klaver TPC, Körmann F, Sukumar PT, da Silva AK, Dutta B, Gutfleisch O, Ponge D, Choi PP, Stephenson LT, Gault B, Raabe D, Li Z (2022) Machine learning-enabled high-entropy alloy discovery. Science 378:78–8510.1126/science.abo4940 ↗
21. Zhu W, Huo W, Wang S, Kurpaska Ł, Fang F, Papanikolaou S, Kim HS, Jiang J (2023) Machine learning-based hardness prediction of high-entropy alloys for laser additive manufacturing. JOM 75:5537–554810.1007/s11837-023-06174-x ↗
22. Creswell A, White T, Dumoulin V, Arulkumaran K, Sengupta B, Bharath AA (2018) Generative adversarial networks: an overview. IEEE Signal Process Mag 35:53–6510.1109/MSP.2017.2765202 ↗
23. Chawla NV, Bowyer KW, Hall LO, Kegelmeyer WP (2002) SMOTE: Synthetic Minority Over-sampling Technique. J Artif Intell Res 16:321–35710.1613/jair.953 ↗
24. M. Arjovsky, S. Chintala, L. Bottou, Wasserstein generative adversarial networks, International conference on machine learning (2017) 214–223.
25. I. Gulrajani, F. Ahmed, M. Arjovsky, V. Dumoulin, A. Courville, Improved training of Wasserstein gans, Advances in neural information processing systems 30 (2017).
26. R. Kruse, S. Mostaghim, C. Borgelt, C. Braune, M. Steinbrecher, Multi-layer perceptrons, Computational intelligence: a methodological introduction (2022) 53–124.10.1007/978-3-030-42227-1_5 ↗
27. S.M. Lundberg, S.I. Lee, A unified approach to interpreting model predictions, Advances in neural information processing systems 30 (2017).
28. Man CK, Quddus M, Theofilatos A, Yu R, Imprialou M (2023) Wasserstein generative adversarial network to address the imbalanced data problem in real-time crash risk prediction. IEEE Trans Intell Transp Syst 23:23002–2301310.1109/TITS.2022.3207798 ↗
29. Guo S, Ng C, Lu J, Liu CT (2011) Effect of valence electron concentration on stability of fcc or bcc phase in high entropy alloys. J Appl Phys 109:10350510.1063/1.3587228 ↗
30. Yang S, Lu J, Xing F, Zhang L, Du Y (2020) Revisit the VEC rule in high entropy alloys (HEAs) with high-throughput CALPHAD approach and its applications for material design-a case study with Al-Co-Cr-Fe-Ni system. Acta Mater 192:11–1910.1016/j.actamat.2020.03.039 ↗
31. Zeng L, Zhan J, Boubeche M, Li K, Li L, Yu P, Wang K, Zhang C, Jin K, Sun Y, Luo H (2023) Superconductivity in the bcc-type high-entropy alloy TiHfNbTaMo. Adv Quantum Technol 6:230021310.1002/qute.202300213 ↗
32. Kube SA, Sohn S, Utz R, Sarkar A, Raabe D (2019) Phase selection motifs in high entropy alloys revealed through combinatorial methods: large atomic size difference favors BCC over FCC. Acta Mater 166:677–68610.1016/j.actamat.2019.01.023 ↗
33. Takeuchi A (2020) Mixing entropy of exact equiatomic high-entropy alloys formed into a single phase. Mater Trans 61:1717–172610.2320/matertrans.MT-M2020141 ↗
34. Zhang Y, Zhou YJ, Lin JP, Chen GL, Liaw PK (2008) Solid-solution phase formation rules for multi-component alloys. Adv Eng Mater 10:534–53810.1002/adem.200700240 ↗
35. Elreedy D, Atiya AF, Kamalov F (2023) A theoretical distribution analysis of synthetic minority oversampling technique (SMOTE) for imbalanced learning. Mach Learn 113:4903–492310.1007/s10994-022-06296-4 ↗
36. Shen Y, Du J, He Y, Wang H, Hao T, Niu X, Jiang Q (2025) AAKAN-WGAN: predicting mechanical properties of magnesium alloys based on generative design and adaptive activation of Kolmogorov-Arnold Networks. Mater Today Commun 47:11319810.1016/j.mtcomm.2025.113198 ↗
37. Alonso JA, Martínez-Lope MJ, Casais MT, García-Muñoz JL, Fernández-Díaz MT, Aranda MAG (2001) High-temperature structural evolution of RNiO3 (R= Ho, Y, Er, Lu) perovskites: charge disproportionation and electronic localization. Phys Rev B 64:09410210.1103/PhysRevB.64.094102 ↗
38. Troparevsky MC, Morris JR, Kent PR, Lupini AR, Stocks GM (2015) Beyond atomic sizes and hume-rothery rules: understanding and predicting high-entropy alloys. JOM 67:2350–236310.1007/s11837-015-1594-2 ↗
39. He QF, Ye YF, Yang Y (2017) Formation of random solid solution in multicomponent alloys: from Hume-Rothery rules to entropic stabilization. J Phase Equilib Diffus 38:416–42510.1007/s11669-017-0560-9 ↗
40. Senkov ON, Wilks GB, Miracle DB, Chuang CP, Liaw PK (2010) Refractory high-entropy alloys. Intermetallics 18:1758–176510.1016/j.intermet.2010.05.014 ↗
41. Bhardwaj V, Zhou Q, Zhang F, Han W, Du Y, Hua K, Wang H (2021) Effect of Al addition on the microstructure and phase constitution of ZrTiNbTa refractory high entropy alloys. Tribol Int 160:10703110.1016/j.triboint.2021.107031 ↗
42. Guo S, Liu CT (2011) Phase stability in high entropy alloys: formation of solid-solution phase or amorphous phase. Prog Nat Sci Mater Int 21:433–44610.1016/S1002-0071(12)60080-X ↗
43. Miracle DB, Senkov ON (2017) A critical review of high entropy alloys and related concepts. Acta Mater 122:448–51110.1016/j.actamat.2016.08.081 ↗
44. Douzas G, Bacao F (2018) Effective data generation for imbalanced learning using conditional generative adversarial networks. Expert Syst Appl 95:464–47110.1016/j.eswa.2017.09.030 ↗
45. C. Bunkhumpornpat, K. Sinapiromsaran, C. Lursinsap, Safe-level-SMOTE: Safe-level-Synthetic Minority Over-Sampling TEchnique for handling the class imbalanced problem, Advances in Knowledge Discovery and Data Mining (2009) 475–482.10.1007/978-3-642-01307-2_43 ↗
46. Lee GC, Li JH, Li ZY (2023) A wasserstein generative adversarial network–gradient penalty-based model with imbalanced data enhancement for network intrusion detection. Appl Sci 13:813210.3390/app13148132 ↗
47. Yu H, Wang QF, Shi JY (2023) Data augmentation generated by generative adversarial network for small sample datasets clustering. Neural Process Lett 55:8365–838410.1007/s11063-023-11315-z ↗
48. Zhang Y, Zuo TT, Tang Z, Gao MC, Dahmen KA, Liaw PK, Lu ZP (2014) Microstructures and properties of high-entropy alloys. Prog Mater Sci 61:1–9310.1016/j.pmatsci.2013.10.001 ↗
49. He J, Ouyang M, Chen Z, Chen D, Liu S (2022) A deep transfer learning fault diagnosis method based on WGAN and minimum singular value for non-homologous bearing. IEEE Trans Instrum Meas 71:350910910.1109/TIM.2022.3160533 ↗