Generative discovery of partial differential equations by learning from math handbooks
File(s)
Published version
Author(s)
Type
Journal Article
Abstract
Data-driven discovery of partial differential equations (PDEs) is a promising approach for uncovering the underlying laws governing complex systems. However, purely data-driven techniques face the dilemma of balancing search space with optimization efficiency. This study introduces a knowledge-guided approach that incorporates existing PDEs documented in a mathematical handbook to facilitate the discovery process. These PDEs are encoded as sentence-like structures composed of operators and basic terms, and used to train a generative model, called EqGPT, which enables the generation of free-form PDEs. A loop of “generation–evaluation–optimization” is constructed to autonomously identify the most suitable PDE. Experimental results demonstrate that this framework can recover a variety of PDE forms with high accuracy and computational efficiency, particularly in cases involving complex temporal derivatives or intricate spatial terms, which are often beyond the reach of conventional methods. The approach also exhibits generalizability to irregular spatial domains and higher dimensional settings. Notably, it succeeds in discovering a previously unreported PDE governing strongly nonlinear surface gravity waves propagating toward breaking, based on real-world experimental data, highlighting its applicability to practical scenarios and its potential to support scientific discovery.
Date Issued
2025-11-21
Date Acceptance
2025-10-08
Citation
Nature Communications, 2025, 16
ISSN
2041-1723
Publisher
Nature Portfolio
Journal / Book Title
Nature Communications
Volume
16
Copyright Statement
© The Author(s) 2025 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Identifier
https://www.ncbi.nlm.nih.gov/pubmed/41271678
PII: 10.1038/s41467-025-65114-2
Publication Status
Published
Coverage Spatial
England
Article Number
10255
Date Publish Online
2025-11-21