Solving Partial Differential Equations in Different Domains by Operator Learning method Based on Boundary Integral Equations

TL;DR

This paper introduces boundary integral equation-based operator learning models, BI-DeepONet and BI-TDONet, capable of solving PDEs on arbitrary domains without retraining, with improved efficiency and handling of oscillatory signals.

Contribution

The paper presents two novel boundary integral equation-based models for PDE solution prediction across arbitrary domains without retraining, incorporating SVD and trigonometric coefficients for enhanced performance.

Findings

Models accurately predict PDE solutions in various domains.

SVD improves the efficiency of BI-TDONet.

Trigonometric coefficients effectively capture oscillatory signals.

Abstract

This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations (BIEs): the Boundary Integral Type Deep Operator Network (BI-DeepONet) and the Boundary Integral Trigonometric Deep Operator Neural Network (BI-TDONet), which are crafted to address PDEs across diverse domains. Once fully trained, these BIE-based models adeptly predict the solutions of PDEs in any domain without the need for additional training. BI-TDONet notably enhances its performance by employing the singular value decomposition (SVD) of bounded linear operators, allowing for the efficient distribution of input functions across its modules. Furthermore, to tackle the issue of function sampling values that do not effectively capture oscillatory and impulse signal characteristics, trigonometric coefficients are utilized as both inputs and outputs in BI-TDONet. Our numerical experiments robustly support and confirm the efficacy of this theoretical framework.