LawMind: A Law-Driven Paradigm for Discovering Analytical Solutions to Partial Differential Equations
This work addresses the challenge of systematically deriving symbolic solutions to PDEs for researchers in physics and applied mathematics, representing a new computational paradigm rather than an incremental improvement.
The authors tackled the problem of deriving closed-form analytical solutions for partial differential equations (PDEs) without relying on data or supervision, and introduced LawMind, a law-driven symbolic discovery framework that successfully recovered solutions for all 100 benchmark PDEs and discovered previously unreported ones.
Partial differential equations (PDEs) encode fundamental physical laws, yet closed-form analytical solutions for many important equations remain unknown and typically require substantial human insight to derive. Existing numerical, physics-informed, and data-driven approaches approximate solutions from data rather than systematically deriving symbolic expressions directly from governing equations. Here we introduce LawMind, a law-driven symbolic discovery framework that autonomously constructs closed-form solutions from PDEs and their associated conditions without relying on data or supervision. By integrating structured symbolic exploration with physics-constrained evaluation, LawMind progressively assembles valid solution components guided solely by governing laws. Evaluated on 100 benchmark PDEs drawn from two authoritative handbooks, LawMind successfully recovers closed-form analytical solutions for all cases. Beyond known solutions, LawMind further discovers previously unreported closed-form solutions to both linear and nonlinear PDEs. These findings establish a computational paradigm in which governing equations alone drive autonomous symbolic discovery, enabling the systematic derivation of analytical PDE solutions.