Predictive Modeling of High-Entropy Alloys and Amorphous Metallic Alloys Using Machine Learning

Abstract

High entropy alloys and amorphous metallic alloys represent two distinct classes of advanced alloy materials, each with unique structural characteristics. Their emergence has garnered considerable interest across the materials science and engineering communities, driven by their promising properties, including exceptional strength. However, their extensive compositional diversity poses substantial challenges for systematic exploration, as traditional experimental approaches and high-throughput calculations struggle to efficiently navigate this vast space. While the recent development in data-driven materials discovery could potentially help, such efforts are hindered by the scarcity of comprehensive data and the lack of robust predictive tools that can effectively link alloy composition with specific properties. To address these challenges, we have deployed a machine-learning-based workflow for feature selection and statistical analysis to afford predictive models that accelerate the data-driven discovery and optimization of these advanced materials. Our methodology is validated through two case studies: (i) a regression analysis of the bulk modulus, and (ii) a classification analysis based on glass-forming ability. The Bayesian-optimized regression model trained for the prediction of bulk modulus achieved an R² of 0.969, an mean absolute error (MAE) of 3.958 GPa, and an root mean square error (RMSE) of 5.411 GPa, while our classification model for predicting glass-forming ability achieved an F1-score of 0.91, an area-under-the-curve of the receiver-operating-characteristic curve of 0.98, and an accuracy of 0.91. Furthermore, by leveraging a wide array of chemical data from diverse literature sources, we have successfully predicted a broad range of properties. This success underscores the efficacy of our modeling approach and emphasizes the importance of a comprehensive feature analysis and judicious feature selection strategy over a mere reliance on complex modeling techniques.

Figure 1

Periodic table highlighting the compositional space represented by the (a) HEA and (b) AMA data sets, where GFA stands for glass-forming ability, D_max the critical casting diameter, CTTs the characteristic transformation temperatures, and EM the elastic moduli.

Figure 2

Overview of our operational workflow as described in Section 2. See ref (54) for a more detailed description. A portion of the figure has been reproduced with permission from ref (54).

Figure 3

Regression of the ML-based predictions of HEA bulk modulus (K) against literature values in the out-of-sample test set and the corresponding distribution of absolute errors. The subfigures (a) and (b) are associated with equimolar quaternary HEAs, whereas (c) and (d) are associated with nonequimolar quaternary HEAs. The dashed red line is drawn to represent the hypothetical case, where the ML-based prediction would equal the literature values. The blue dot-dash line is a linear fit generated using the OLS method. Within the error distribution plots, the red dotted line symbolizes the MAE, and the orange dashed line denotes the RMSE.

Figure 4

GBFS results for the prediction of K values in equimolar quaternary HEAs. The figure shows the performance of GBDTs on (a) the training set and (b) the validation set, where regression models are trained recursively with an increasing subset of features, beginning from the most relevant feature based on the realized total loss reduction. (c) Multicollinearity reduction for the regression analysis, showing the dendrogram of the hierarchical agglomerative clustering using the remaining 59 features after performing the correlation analysis. The dashed horizontal line in black represents the distance threshold of 1.5 unit of Ward’s linkage distance.

Figure 5

(a) Feature relevance and (b) permutation-based feature importance plots for the regression analysis of K in equimolar quaternary HEAs, displaying the five most significant features.

Figure 6

Results based on the SHAP framework: (a) the average contribution (i.e., the mean absolute SHAP value) of five features that are identified as having the greatest contributions to the model output. A positive SHAP value indicates a positive contribution to the prediction of K values. (b) The beeswarm plot illustrates the impact of these features on the model output by plotting each instance as a single data point together with the SHAP value on the x-axis, where the y-axis is consistent with (a). The color scheme corresponds to the original feature value and the broadening shows the density of instances (cf. the density plot).

Figure 7

Property predictions for HEAs. The dashed red line is drawn to represent the hypothetical case, where the ML-based prediction would equal the literature values. The blue dot-dash lines represent linear fits generated using the OLS method, applied to various materials properties: (a) elastic constant C₁₁, (b) elastic constant C₁₂, (c) elastic constant C₄₄, (d) Young’s modulus, (e) shear modulus, (f) Wigner–Seitz radius, (g) formation enthalpy, (h) total energy, (i) Zener anisotropy, (j) Pugh ratio, (k) Poisson ratio, and (l) universal anisotropy.

Figure 8

Confusion matrix for the classification of alloys by their glass-forming ability.

Figure 9

GBFS results for the prediction of glass-forming ability of AMAs. The figure shows the performance of GBDTs on (a) the training set and (b) the validation set, where classification models are trained recursively with an increasing subset of features, beginning from the most relevant feature based on the realized total loss reduction. (c) Multicollinearity reduction for the classification analysis, showing the dendrogram of the hierarchical agglomerative clustering using the remaining 64 features after performing the correlation analysis. The dashed horizontal line in black represents the distance threshold of 1.5 unit of Ward’s linkage distance.

Figure 10

(a) Feature relevance and (b) permutation-based feature importance plots for the classification analysis of alloys by their glass-forming ability, displaying the five most significant features.

Figure 11

Property predictions of AMAs. The dashed red line is drawn to represent the hypothetical case, where the ML-based prediction would equal the literature values. The blue dot-dash lines are linear fits generated using OLS method, applied to various materials properties: (a) glass transition temperature T_g, (b) onset of crystallization temperature T_x, (c) liquidus temperature T_l, (d) critical casting diameter D_max, (e) bulk modulus, and (f) shear modulus.