Solving Partial Differential Equations in Different Domains by Operator Learning method Based on Boundary Integral Equations
This addresses the problem of domain generalization in PDE solving for computational science and engineering, representing an incremental improvement over existing operator learning methods.
The paper tackles solving partial differential equations (PDEs) on arbitrary domains without retraining by introducing two operator learning models based on boundary integral equations, with BI-TDONet showing enhanced performance through singular value decomposition and trigonometric coefficients.
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.