Physics-Constrained Symbolic Regression via Reinforcement Learning: A Rigorous Framework for Discovering Closed-Form Solutions to Differential Equations

Abstract

Discovering exact, interpretable, closed-form solutions to differential equations (DEs) remains one of the most intellectually demanding challenges at the intersection of applied mathematics and computational intelligence. Classical analytical techniques, while rigorous, are largely restricted to well-structured linear systems and fail to generalise to the nonlinear, high-dimensional configurations that dominate contemporary science and engineering. The present study introduces and theoretically and empirically validates a physics-constrained symbolic regression framework, designated PCSRL (Physics-Constrained Symbolic Regression via Reinforcement Learning), that unifies policy-gradient reinforcement learning with an exact physical constraint evaluator to recover closed-form symbolic solutions for a broad class of ordinary and partial differential equations. Rigorous evaluation across six canonical benchmark problems including linear and nonlinear Poisson, heat, and wave equations in both two- and three-dimensional spatial domains demonstrates that PCSRL achieves complete symbolic recovery (recovery rate = 100%) and physical-constraint residuals multiple orders of magnitude below those of competing methods, including genetic programming-based symbolic regression, Kolmogorov–Arnold Networks, and PINN-assisted symbolic regression pipelines. These results establish a principled, reproducible methodology for the machine discovery of physically exact symbolic solutions, with direct implications for mathematical physics, computational fluid dynamics, and scientific machine learning. Practically, the framework enables engineers and scientists to obtain transparent, analytically tractable models for complex physical systems, facilitating rapid design optimisation, stability analysis, and knowledge discovery in domains where black-box numerical approximations are insufficient.