LawMind: A Law-Driven Paradigm for Discovering Analytical Solutions to Partial Differential Equations
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arXiv
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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912967132446720 |
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| author | Zheng, Min-Yi Zhang, Shengqi Wu, Liancheng Zhong, Jinghui Chen, Shiyi Ong, Yew-Soon |
| author_facet | Zheng, Min-Yi Zhang, Shengqi Wu, Liancheng Zhong, Jinghui Chen, Shiyi Ong, Yew-Soon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14353 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | LawMind: A Law-Driven Paradigm for Discovering Analytical Solutions to Partial Differential Equations Zheng, Min-Yi Zhang, Shengqi Wu, Liancheng Zhong, Jinghui Chen, Shiyi Ong, Yew-Soon Symbolic Computation 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. |
| title | LawMind: A Law-Driven Paradigm for Discovering Analytical Solutions to Partial Differential Equations |
| topic | Symbolic Computation |
| url | https://arxiv.org/abs/2603.14353 |