Component Fourier Neural Operator for Singularly Perturbed Differential Equations

TL;DR

This paper introduces ComFNO, a novel neural operator that enhances the solution of singularly perturbed differential equations by integrating asymptotic analysis, leading to improved accuracy and adaptability over existing methods.

Contribution

The paper proposes ComFNO, an innovative neural operator that incorporates prior asymptotic knowledge, improving the solution accuracy and generalization for SPDEs compared to standard FNO.

Findings

ComFNO outperforms vanilla FNO in accuracy across various SPDEs.

ComFNO demonstrates strong adaptability to different data distributions.

ComFNO performs well in few-shot learning scenarios.

Abstract

Solving Singularly Perturbed Differential Equations (SPDEs) poses computational challenges arising from the rapid transitions in their solutions within thin regions. The effectiveness of deep learning in addressing differential equations motivates us to employ these methods for solving SPDEs. In this manuscript, we introduce Component Fourier Neural Operator (ComFNO), an innovative operator learning method that builds upon Fourier Neural Operator (FNO), while simultaneously incorporating valuable prior knowledge obtained from asymptotic analysis. Our approach is not limited to FNO and can be applied to other neural network frameworks, such as Deep Operator Network (DeepONet), leading to potential similar SPDEs solvers. Experimental results across diverse classes of SPDEs demonstrate that ComFNO significantly improves accuracy compared to vanilla FNO. Furthermore, ComFNO exhibits natural adaptability to diverse data distributions and performs well in few-shot scenarios, showcasing its excellent generalization ability in practical situations.