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

2025-10-25

AI Summary

Solving high-frequency oscillatory partial differential equations (PDEs) on complex geometric domains remains challenging: conventional numerical methods suffer from prohibitive computational cost, while deep learning approaches lack interpretability and suffer from optimization difficulties. To address this, we propose the multiscale Finite Expression (FEX) method—a novel symbolic-numerical framework integrating a symbolic spectral composition module for explicit multiscale oscillatory modeling, an enhanced linear input layer to boost representational capacity, and an eigenvalue-constrained loss to enforce physical consistency. FEX is the first method enabling closed-form, interpretable, and high-accuracy solutions for both oscillatory and eigenvalue-type PDEs. On complex domains with heterogeneous-shaped holes, FEX significantly outperforms state-of-the-art neural networks and traditional solvers in accuracy, efficiency, and analytical interpretability—establishing a new paradigm for oscillatory PDE solution.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Learning to Search
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Solving oscillatory PDEs on complex domains efficiently
• Overcoming expensive representations in traditional numerical methods
• Improving accuracy and interpretability of PDE solutions

Innovation

• Symbolic spectral composition captures multiscale oscillatory behavior
• Redesigned linear input layer expands model expressivity
• Eigenvalue formulation extends method to eigenvalue PDEs