Finite Expression Methods for Discovering Physical Laws from Data
This work addresses the problem of discovering physical laws from data for researchers in scientific and engineering fields, representing a novel method for a known bottleneck rather than an incremental improvement.
The paper tackles the challenge of deriving analytical expressions for nonlinear dynamics from limited data by introducing the finite expression method (FEX), a deep symbolic learning approach that generates governing equations from observed dynamic data, and demonstrates that FEX outperforms existing methods like PDE-Net and SINDy in numerical performance across various problems.
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.