Physics-guided neural network with adaptive constraint weighting for predicting the bulk modulus of high-entropy alloys

High-entropy alloys (HEAs) present significant challenges for property prediction due to their vast compositional freedom and limited reliable data. Without tractable governing equations, elastic properties have been rationalized using physically motivated descriptors such as elastic bounds and empirical parameters. While incorporating such constraints improves robustness, predictive performance remains sensitive to how multiple constraints are weighted. Here, we propose a physics-guided neural network integrating four complementary constraints: Voigt and Reuss elastic bounds from classical elasticity theory, valence electron concentration (VEC) for phase stability, and atomic size mismatch (δ) reflecting lattice distortion. These are incorporated as soft regularization terms, automatically balanced via Lagrangian-dual adaptive weighting. The framework was evaluated on 1,117 HEAs with bulk moduli ranging from 12.5 to 428.7 GPa. Four-fold cross-validation demonstrated robust convergence with coefficients of variation below 7% for all constraint weights. Compared with an unconstrained neural network (MAE: 5.583 GPa, R²: 0.8657) and a fixed-weight physics-constrained model (MAE: 5.240 GPa, R²: 0.8526), the proposed framework achieved best performance (MAE: 4.573 GPa, R²: 0.9002), corresponding to MAE reductions of 18.1% and 12.7%, respectively. These results demonstrate that combining physically motivated constraints with automatic λ tuning enhances extrapolation performance while maintaining consistency with fundamental mechanical bounds. The approach provides a physically interpretable and data-efficient framework for accelerating materials design in high-dimensional alloy systems under limited data conditions. • Physics-constrained neural network predicts bulk modulus of HEAs from composition. • Adaptive constraint weighting balances data fit with physics-based constraints. • Model outperforms existing data-driven methods in bulk modulus prediction accuracy. • Framework enables reliable screening of candidate HEAs across composition space.