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arXiv · 2603.14353

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

Min-Yi Zheng·Shengqi Zhang·Liancheng Wu·Jinghui Zhong·Shiyi Chen·Yew-Soon Ong
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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.

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Min-Yi Zheng, Shengqi Zhang, Liancheng Wu, Jinghui Zhong, Shiyi Chen, Yew-Soon Ong. 2026-03-15. LawMind: A Law-Driven Paradigm for Discovering Analytical Solutions to Partial Differential Equations. https://arxiv.org/abs/2603.14353

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