Machine learning-driven inverse design of high-hardness multi-principal element alloys through symbolic regression

Conventional trial-and-error metallurgical approaches fail to efficiently develop high-hardness multi-principal element alloys (MPEAs), limited by poor physical interpretability of existing machine learning (ML) hardness prediction models and the lack of targeted inverse composition optimization. This work establishes a closed-loop physics-informed ML framework integrating symbolic regression (SR), CatBoost surrogate modelling and multi-metaheuristic optimization to address the two core bottlenecks. A comprehensive descriptor library is constructed by integrating elemental fractions, pymatgen-derived atomic physicochemical properties and classical Hume-Rothery metallurgical parameters, which quantitatively captures lattice distortion, electronic effects and phase transformation mechanisms governing alloy hardness. Genetic programming-based symbolic regression automatically extracts three nonlinear descriptors that encode standardized physicochemical variables, reducing the original 39-dimensional feature space by 33% and remarkably elevating predictive accuracy. The SR-enhanced CatBoost model achieves a test-set coefficient of determination R 2 = 0.885 and RMSE = 77.2 HV, far outperforming traditional semi-empirical formulas and baseline ML models. SHAP analysis verifies that the three SR features contribute 52% of total predictive weight, explicitly elucidating non-monotonic hardening responses including hardness saturation under excessive atomic size mismatch and the trade-off between bulk modulus and dislocation storage capacity. A triangulated inverse design pipeline combining genetic algorithm, particle swarm optimization and whale optimization algorithm is deployed to maximize Vickers hardness under four- and five-principal-element constraints. Cross-algorithm convergence shows stability of the search under a shared surrogate rather than global optimality. The 22 candidates that clear the physical screen and the domain test span 716.6-783.1 HV. Refractory W and Mo act as primary strengthening elements, while moderate Al addition enables simultaneous lightweighting and possible precipitation hardening. This integrated framework unifies interpretable forward hardness prediction, mechanistic analysis and targeted compositional inverse design, drastically cutting experimental screening costs. Future research will incorporate processing and microstructural variables to realize multi-objective optimization balancing hardness, density and ductility.