Physics-informed machine learning framework integrating solid solution strengthening theory for accelerated hardness prediction in high-entropy alloys

High-Entropy Alloys (HEAs) exhibit exceptional stability in extreme environments, yet their expansive design space presents a “curse of dimensionality” for traditional discovery methods. While Machine Learning (ML) offers a data-driven paradigm for material screening, the scarcity of experimental data often results in overfitting and limited physical interpretability. To address these challenges, this study proposes a Hybrid Physics-Informed Machine Learning (Hybrid PIML) framework for accelerated hardness prediction. By integrating classical solid solution strengthening theory with a Residual Learning Artificial Neural Network (ANN), the model explicitly embeds the physical coupling of shear modulus and lattice distortion (G ⋅ δ_r^2⁄3) as prior knowledge. This approach ensures predictions adhere to metallurgical principles while significantly outperforming benchmark algorithms, achieving a coefficient of determination (R²) of 0.976 and reducing the Root Mean Square Error (RMSE) by approximately 43%. SHapley Additive exPlanations (SHAP) analysis confirms that physics-enhanced features dominate the decision-making process, validating the model's internalization of strengthening mechanisms. Furthermore, the research elucidates a phase-dependent non-linear correlation between hardness and yield strength, correcting the failure of the classical Tabor formula in work-hardening Face-Centered Cubic (FCC) alloys. Finally, a high-throughput virtual screening funnel based on this framework successfully identified optimized non-equiatomic candidates within the refractory Co-Cr-Ti-Mo-W system. This work establishes a precise, physically consistent pathway for inverse material design under data-constrained conditions.