Symbolic regression guided by interpretable machine learning for formulating a fracture toughness law from micro-indentation data

Abstract

Although artificial intelligence (AI) plays a crucial role in material property prediction, its models often lack interpretability, and the physical mapping relationships behind their excellent performance are seldom addressed. This work introduces a systematic framework designed to remove redundancy from high-dimensional features and discover underlying mathematical laws. Specifically, we demonstrate this framework by deriving an explicit fracture toughness law from micro-indentation data. This framework, centered on Symbolic Regression (SR), leverages the SHapley Additive exPlanations (SHAP) technique to analyze a trained Artificial Neural Network (ANN). The ANN is first trained on mechanical parameters, including elastic–plastic and fracture properties. Subsequently, the SHAP technique is integrated to quantify feature importance and reduce dimensionality, thereby laying the groundwork for SR to uncover the physical equation. The results show that the ANN model’s generalization capability is improved by removing certain features based on their ranked importance. Three dominant features, namely indentation modulus (E_IT), hardness (H_IT), and creep deformation (H_creep), are identified as key predictors and subsequently fed into a symbolic regression model to investigate the potential functional relationship between elastic–plastic properties and fracture toughness. After balancing model accuracy and complexity, the nonlinear relationship between elastic–plastic properties and fracture toughness is described by an SR-generated mathematical equation. In total, this work bridges data-driven modeling with classical fracture mechanics, offering a compact, explainable relationship for estimating fracture toughness in complex geomaterials.