Finite Expression Methods for Discovering Physical Laws from Data

TL;DR

This paper introduces the finite expression method (FEX), a deep symbolic learning approach that discovers analytical governing equations from limited nonlinear dynamic data, outperforming existing methods in accuracy and flexibility.

Contribution

The paper presents a novel FEX approach that effectively learns finite-structure analytical expressions for PDEs and dynamical systems from data, advancing symbolic discovery techniques.

Findings

FEX outperforms PDE-Net, SINDy, GP, and SPL in numerical accuracy.

FEX accurately models time-dependent PDEs and nonlinear systems.

FEX demonstrates high flexibility and expressive power in symbolic approximation.

Abstract

Nonlinear dynamics is a pervasive phenomenon observed in scientific and engineering disciplines. However, the task of deriving analytical expressions to describe nonlinear dynamics from limited data remains challenging. In this paper, we shall present a novel deep symbolic learning method called the "finite expression method" (FEX) to discover governing equations within a function space containing a finite set of analytic expressions, based on observed dynamic data. The key concept is to employ FEX to generate analytical expressions of the governing equations by learning the derivatives of partial differential equation (PDE) solutions through convolutions. Our numerical results demonstrate that our FEX surpasses other existing methods (such as PDE-Net, SINDy, GP, and SPL) in terms of numerical performance across a range of problems, including time-dependent PDE problems and nonlinear dynamical systems with time-varying coefficients. Moreover, the results highlight FEX's flexibility and expressive power in accurately approximating symbolic governing equations.