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

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.

Figures (3)

• Figure 1: LawMind. a, Given a governing law and associated conditions, LawMind discovers analytical solutions without relying on data or external supervision. The example illustrates the recovery of an analytical solution to a heat equation $u_{t} - au_{x,x} = 0$ under prescribed initial conditions $u(x,0) = \exp(x)$. b-e, Demonstration of the law-driven symbolic discovery process, including stage-wise activation of variables into the terminal set to incrementally control the symbolic search space (b), assembly of candidate subtrees from these sets to serve as building blocks for expression expansion (c), evolutionary expansion of the current expression by inserting assembled subtrees at designated positions (d), and law-driven evaluation via governing equations and conditions (e). f, Overview of the complete LawMind workflow.
• Figure 2: Benchmark design and recovery of analytical solutions across five PDE categories. a, The 100-problem benchmark is curated from two authoritative reference handbooks polyanin2001handbookpolyanin2003handbook covering linear and nonlinear PDEs, ensuring broad mathematical coverage. b, Five representative benchmark problems, one from each PDE category: first-order PDEs, second-order linear parabolic PDEs, second-order linear hyperbolic PDEs, second-order nonlinear parabolic PDEs, and second-order nonlinear hyperbolic PDEs. Each problem is presented as a card front specifying the governing PDE and initial condition (IC). c, The corresponding card backs show the closed-form analytical solutions recovered by LawMind. Symbolic verification confirms exact satisfaction of the governing equation and initial condition. Status indicates whether each solution is previously known or newly discovered.
• Figure 4: Closed-form solutions beyond classical handbooks. a, b, Two newly discovered analytical solutions to a third-order linear PDE under distinct initial conditions. c, d, Two newly discovered analytical solutions to a third-order nonlinear PDE under distinct initial conditions. For each case, the card front presents the governing PDE and initial condition, and the card back presents the closed-form solution recovered by LawMind.