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.

翻译：非线性动力学是科学与工程领域中普遍存在的现象。然而，从有限数据中推导描述非线性动力学的解析表达式仍然是一项挑战。本文提出一种新颖的深度符号学习方法——"有限表达式方法"（FEX），该方法基于观测的动态数据，在包含有限解析表达式的函数空间中，用于发现控制方程。其核心思想是通过卷积学习偏微分方程（PDE）解的导数，利用FEX生成控制方程的解析表达式。数值结果表明，在时间相关PDE问题及含时变系数的非线性动力系统等一系列问题中，我们的FEX在数值性能上超越了现有方法（如PDE-Net、SINDy、GP和SPL）。此外，结果还突出了FEX在精确近似符号控制方程方面的灵活性与表达能力。