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.

翻译：[translated abstract in Chinese] 偏微分方程（PDE）编码了基本的物理规律，然而许多重要方程的闭式解析解仍然未知，且通常需要大量的人类洞察力才能推导出来。现有的数值方法、物理信息方法和数据驱动方法从数据中近似求解，而不是直接从控制方程中系统性地推导符号表达式。在此，我们引入LawMind，一种法律驱动的符号发现框架，它能够从偏微分方程及其相关条件自主构建闭式解，无需依赖数据或监督。通过将结构化符号探索与物理约束评估相结合，LawMind仅在控制规律的指导下逐步组装有效的解分量。在从两本权威手册中选取的100个基准偏微分方程上评估，LawMind成功恢复了所有情况的闭式解析解。除了已知的解，LawMind还进一步发现了线性和非线性偏微分方程之前未报道的闭式解。这些发现建立了一种计算范式，其中仅控制方程就能驱动自主符号发现，从而能够系统地推导偏微分方程的解析解。