Solving partial differential equations (PDEs) with highly oscillatory solutions on complex domains remains a challenging and important problem. High-frequency oscillations and intricate geometries often result in prohibitively expensive representations for traditional numerical methods and lead to difficult optimization landscapes for machine learning-based approaches. In this work, we introduce an enhanced Finite Expression Method (FEX) designed to address these challenges with improved accuracy, interpretability, and computational efficiency. The proposed framework incorporates three key innovations: a symbolic spectral composition module that enables FEX to learn and represent multiscale oscillatory behavior; a redesigned linear input layer that significantly expands the expressivity of the model; and an eigenvalue formulation that extends FEX to a new class of problems involving eigenvalue PDEs. Through extensive numerical experiments, we demonstrate that FEX accurately resolves oscillatory PDEs on domains containing multiple holes of varying shapes and sizes. Compared with existing neural network-based solvers, FEX achieves substantially higher accuracy while yielding interpretable, closed-form solutions that expose the underlying structure of the problem. These advantages, often absent in conventional finite element, finite difference, and black-box neural approaches, highlight FEX as a powerful and transparent framework for solving complex PDEs.

翻译：在复杂域上求解具有高度振荡解的偏微分方程（PDEs）仍然是一个重要且具有挑战性的问题。高频振荡和复杂几何结构通常导致传统数值方法需要极其昂贵的表示成本，并为基于机器学习的方法带来困难的优化地形。本文提出一种增强型有限表达式方法（FEX），旨在以更高的精度、可解释性和计算效率应对这些挑战。该框架包含三项关键创新：一个符号谱组合模块，使FEX能够学习和表示多尺度振荡行为；一个重新设计的线性输入层，显著扩展了模型的表达能力；以及一个特征值公式，将FEX扩展至涉及特征值偏微分方程的新问题类别。通过大量数值实验，我们证明FEX能够精确求解包含多种形状和尺寸孔洞的复杂域上的振荡偏微分方程。与现有的基于神经网络的求解器相比，FEX在获得更高精度的同时，还能产生可解释的闭式解，从而揭示问题的底层结构。这些优势在传统的有限元法、有限差分法以及黑箱神经网络方法中通常难以实现，凸显了FEX作为求解复杂偏微分方程的一种强大且透明的框架价值。