Accelerated Design of Mechanically Hard Magnetically Soft High-entropy Alloys via Multi-objective Bayesian Optimization

Abstract

Designing high-entropy alloys (HEAs) that are both mechanically hard and possess soft magnetic properties is inherently challenging, as a trade-off is needed for mechanical and magnetic properties. In this study, we optimize HEA compositions using a multi-objective Bayesian optimization (MOBO) framework to achieve simultaneous optimal mechanical and magnetic properties. An ensemble surrogate model is constructed to enhance the accuracy of machine learning surrogate models, while an efficient sampling strategy combining Monte Carlo sampling and acquisition function is applied to explore the high-dimensional compositional space. The implemented MOBO strategy successfully identifies Pareto-optimal compositions with enhanced mechanical and magnetic properties. The ensemble model provides robust and reliable predictions, and the sampling approach reduces the likelihood of entrapment in local optima. Our findings highlight specific elemental combinations that meet the dual design objectives, offering guidance for the synthesis of next-generation HEAs.

Figures (4)

• Figure 1: Schematic overview of the multi-objective Bayesian optimization (MOBO) framework for high-entropy alloy (HEA) design. The workflow integrates four key components: (i) Design space, where candidate alloys are generated from a 10-element chemical pool with nonequimolar compositions and different crystal structures; (ii) Quantum mechanical method, where the coherent potential approximation treats chemical disorder and enables evaluation of magnetic and mechanical properties; (iii) Feature engineering, where datasets from quantum calculations are transformed into descriptors for machine learning; (iv) Machine learning (ML) surrogate model (e.g., multilayer perceptrons (MLPs), random forests (RFs), and gradient-boosted decision trees (LightGBM)), where ensemble learning with bootstrapping and stacking provides predictive accuracy and uncertainty quantification. These ML models are embedded into the MOBO loop, balancing exploration and exploitation to iteratively suggest new HEA candidates until convergence.
• Figure 2: Convergence of the multi-objective search and per-property learning dynamics. (a) Best-so-far Pareto-front indicator (blue, left axis) increases monotonically with iteration, while red box-and-whisker plots report the distribution of attained hypervolume (right axis) across evaluations in each iteration, evidencing a rapid expansion of the dominated objective space followed by saturation. (b–e) Iteration-wise trajectories of the four target properties: total magnetic moment $M_{\mathrm{Tot}} [\mu_B]$, Curie temperature $T_C$ [K], Pugh's ratio $B/G$ and Cauchy pressure $C_{12}$ - $C_{44}$ [GPa]. Dots denote the per-iteration mean over evaluated candidates; shaded envelopes indicate the confidence interval computed from the samples in each iteration.
• Figure 3: Pairwise objective landscapes obtained from multi-objective Bayesian optimization of high-entropy alloys. Each panel (a–f) shows the joint distribution of candidate alloys across two objectives: (a) Curie temperature $T_{\mathrm{C}}$ vs. total magnetic moment $M_{\mathrm{Tot}}$, (b) Pugh’s ratio $B/G$ vs. $M_{\mathrm{Tot}}$, (c) Cauchy pressure $C_{12}-C_{44}$ vs. $M_{\mathrm{Tot}}$, (d) $B/G$ vs. $T_{\mathrm{C}}$, (e) $C_{12}-C_{44}$ vs. $T_{\mathrm{C}}$, and (f) $C_{12}-C_{44}$ vs. $B/G$. Blue dots denote sampled candidates, shaded contours indicate density levels, and black star markers highlight Pareto-optimal solutions. Red dashed lines represent reference thresholds used to delineate desirable property regimes.
• Figure 4: Evolution of composition and feature importance. (a) Averaged elemental compositions of candidate alloys over consecutive groups of ten optimization iterations, shown as stacked bar plots with color-coded contributions from the ten constituent elements (Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn). Each bar represents the normalized mean element ratio within one batch of ten evaluated compositions. (b) Normalized feature importance of selected elemental descriptors with respect to the four optimization objectives: saturation magnetization $M_{s}$, Curie temperature $T_{C}$, Pugh’s ratio $B/G$, and Cauchy pressure $C_{12}-C_{44}$.