Generative Discovery of Partial Differential Equations by Learning from Math Handbooks
This work addresses the challenge of balancing search space and optimization efficiency in PDE discovery for complex systems, offering a method that is incremental by building on existing knowledge to enhance scientific discovery in practical scenarios.
The study tackled the problem of data-driven discovery of partial differential equations (PDEs) by introducing a knowledge-guided approach that uses existing PDEs from a mathematical handbook to train a generative model, enabling the recovery of various PDE forms with high accuracy and computational efficiency, including a previously unreported PDE for strongly nonlinear surface gravity waves.
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.