Nonlinear dynamics is a pervasive phenomenon observed in various scientific and engineering disciplines. However, uncovering analytical expressions that describe nonlinear dynamics from limited data remains a challenging and essential task. In this paper, we propose a new deep symbolic learning method called the ``finite expression method'' (FEX) to identify the governing equations within the space of functions containing a finite set of analytic expressions, based on observed dynamic data. The core idea is to leverage FEX to generate analytical expressions of the governing equations by learning the derivatives of partial differential equation (PDE) solutions using convolutions. Our numerical results demonstrate that FEX outperforms all existing methods (such as PDE-Net, SINDy, GP, and SPL) in terms of numerical performance across various problems, including time-dependent PDE problems and nonlinear dynamical systems with time-varying coefficients. Furthermore, the results highlight that FEX exhibits flexibility and expressive power in accurately approximating symbolic governing equations, while maintaining low memory and favorable time complexity.

翻译：非线性动力学是众多科学与工程学科中普遍存在的现象。然而，从有限数据中揭示描述非线性动力学的解析表达式仍是一项具有挑战性的关键任务。本文提出一种全新的深度符号学习方法——"有限表达式方法"（FEX），旨在基于观测到的动态数据，在包含有限解析表达式集合的函数空间中识别控制方程。其核心思想是利用FEX通过卷积学习偏微分方程（PDE）解的导数，从而生成控制方程的解析表达式。数值结果表明，在各类问题（包括含时偏微分方程问题和具有时变系数的非线性动力系统）中，FEX在数值性能上均优于现有方法（如PDE-Net、SINDy、GP和SPL）。此外，结果还强调，FEX在保持低内存占用和良好时间复杂度的同时，在精确逼近符号化控制方程方面展现出灵活性和表达能力。