LawMind: A Law-Driven Paradigm for Discovering Analytical Solutions to Partial Differential Equations

TL;DR

LawMind is a novel framework that autonomously derives closed-form solutions to PDEs directly from governing laws, successfully solving all benchmark cases and discovering new solutions without data or supervision.

Contribution

It introduces a law-driven symbolic discovery method that systematically constructs analytical solutions to PDEs solely from their equations and conditions, without relying on data.

Findings

Successfully solves all 100 benchmark PDEs with known solutions.

Discovers new closed-form solutions for linear and nonlinear PDEs.

Operates without data or supervision, driven solely by governing laws.

Abstract

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.