Generative discovery of partial differential equations by learning from math handbooks.

Area of Science:

• Computational Physics
• Applied Mathematics
• Scientific Machine Learning

Background:

• Data-driven discovery of partial differential equations (PDEs) is crucial for understanding complex systems.
• Purely data-driven methods struggle with balancing search space and optimization efficiency.
• Incorporating existing knowledge can enhance PDE discovery.

Purpose of the Study:

• To develop a knowledge-guided approach for discovering partial differential equations (PDEs).
• To improve the efficiency and accuracy of data-driven PDE discovery, especially for complex equations.
• To demonstrate the framework's applicability to real-world scientific problems.

Main Methods:

• A knowledge-guided framework using a generative model (EqGPT) trained on handbook PDEs.
• Encoding PDEs as sentence-like structures of operators and terms.
• Implementing a "generation-evaluation-optimization" loop for autonomous PDE identification.

Main Results:

• The framework accurately and efficiently recovers various PDE forms, including those with complex temporal or spatial terms.
• Demonstrated generalizability to irregular spatial domains and higher dimensions.
• Successfully discovered a novel PDE for strongly nonlinear surface gravity waves from experimental data.

Conclusions:

• The knowledge-guided approach significantly enhances data-driven PDE discovery.
• EqGPT offers a powerful tool for uncovering complex physical laws and supports practical scientific discovery.
• The method shows promise for tackling challenging problems in various scientific domains.