Physics-informed neural networks and symbolic regression for equation discovery in non-destructive evaluation of composite plates

Data-driven frameworks, such as deep neural networks, face significant challenges in data acquisition for the non-destructive evaluation of composite materials, particularly when dense measurement points and high accuracy are required. Furthermore, the low interpretability of neural networks limits their usability and reduces user confidence. To address these issues, this study incorporates a novel physics-informed loss term—a frequency prediction loss—into the PINN framework, together with a generalized loss normalization strategy. These additions embed prior physical knowledge into the network and enhance its predictive capability. When applied to sparse frequency-series datasets generated by finite element method, our 1D-PINN achieves an average prediction error of 7.23% and improves performance by 13.10% compared to networks without physical constraints, illustrating the efficacy of the proposed loss terms in enhancing model generalization. To enhance interpretability, we combine the gradient-based saliency mapping technique with symbolic regression (SR) to derive empirical formulas that accurately relate key natural frequencies to mechanical properties. This approach achieves an average prediction error of less than 4.33% when predicting the longitudinal modulus of elasticity. In comparison, conventional networks struggle to generalize effectively due to insufficient focus on key input features. Additionally, 1D-PINN and the derived empirical formulas enable accurate, non-destructive monitoring of progressive damage in composite laminates, achieving average prediction errors of 2.94% and 1.99%, respectively. These findings demonstrate the high generalizability and accuracy of the PINN-SR method, making it well-suited for health monitoring of engineering structures.