Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domains

TL;DR

This paper introduces an enhanced Finite Expression Method (FEX) that effectively solves oscillatory PDEs on complex domains by combining spectral composition, expanded expressivity, and eigenvalue formulations, outperforming traditional methods.

Contribution

The paper presents a novel FEX framework with spectral composition, a redesigned input layer, and eigenvalue capabilities, improving accuracy, interpretability, and efficiency for complex oscillatory PDEs.

Findings

FEX accurately solves oscillatory PDEs on complex domains.

FEX achieves higher accuracy than neural network-based solvers.

FEX provides interpretable, closed-form solutions exposing problem structure.

Abstract

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.