Finite Expression Methods for Discovering Physical Laws from Data

TL;DR

The paper tackles the problem of extracting analytical governing equations from data for nonlinear PDEs and dynamical systems. It introduces the Finite Expression Method (FEX), which searches within a finite operator space using binary-tree expressions, learns derivatives via trainable convolution kernels, and selects operator sequences with reinforcement learning while tuning leaf parameters through continuous optimization; the approach uses the reward R(e)=(1+L(e))^{-1} and combines first- and second-order optimizers for efficiency. Across Burgers equations (constant and varying coefficients), the 2D Hopf normal form, and JMAK growth, FEX consistently outperforms PDE-Net 2.0, SINDy, GP, and SPL in term recovery and accuracy, including capturing time-varying coefficients symbolic-ly. The method yields interpretable, compact symbolic expressions that generalize to higher dimensions and complex nonlinearities, offering a practical tool for data-driven discovery of physical laws in physics and engineering.

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.

Figures (10)

• Figure 1: Overview of different symbolic methods for discovering governing equations from data.
• Figure 2: The representation of the major components of our FEX. (a) The searching loop for the symbolic solution consists of expression generation, score computation, controller update, and candidate optimization. (b) The depiction of the expression generation with a binary tree and a controller $\chi$. The appropriate derivatives of $u$ at the leaf level of the binary tree will be learned via continuous optimization.
• Figure 3: The binary tree structure in the fully combinatorial optimization scheme. Only first-order differential operators are included in the dictionary of unary operators. Given a prescribed dictionary of unary and binary operators, e.g., $\{+,-,\times,/, \sin, \cos, \exp, \partial_x,\partial_y,\dots\}$, when the inputs are chosen as a function $u$ and its variables, complex governing equations of $u$ with possibly high-order derivatives and varying coefficients can be generated via operator compositions through a deep binary tree.
• Figure 4: The binary tree structure in the partially relaxed combinatorial optimization for learning derivatives. Instead of including differential operators in the operator dictionary, leaf nodes are designed to learn to select appropriate derivatives of $u$ to form a governing equation via continuous optimization.
• Figure 5: Left: Worst results among 5 runs of three methods. Right: Best results among 5 runs of three methods