Interpretable Physics-Informed Neural Networks for Implicit Physics Recovery

Abstract

The integration of prior physical knowledge into deep learning architectures has revolutionized the field of computational science, particularly through the advent of Physics-Informed Neural Networks. Despite their empirical success in solving forward and inverse problems governed by partial differential equations, standard Physics-Informed Neural Networks often function as black-box models, severely limiting their applicability in discovering unknown or implicit physical laws from noisy observational data. This paper presents a comprehensive investigation into Interpretable Physics-Informed Neural Networks designed specifically for implicit physics recovery. By integrating symbolic regression and sparse feature selection mechanisms directly into the loss landscape of the neural network, the proposed framework not only approximates the solution to the governing equations but also yields explicit, human-readable mathematical expressions of the underlying physical processes. We conduct a rigorous theoretical and empirical analysis across various dynamical systems to demonstrate how the inclusion of interpretability constraints enhances the robustness, generalization, and physical consistency of the model. Furthermore, we address the critical challenge of balancing data-driven flexibility with physical law adherence. Through extensive numerical experiments on both synthetic and real-world observational datasets, we show that the interpretable framework accurately recovers hidden differential operators even in the high-noise regime, outperforming conventional architectures. The findings underscore the necessity of transparent machine learning models in scientific computing and provide a robust foundation for future research in automated scientific discovery.