Three operator learning models for solving boundary integral equations in 2D connected domains

Abstract

This paper proposes three operator learning models based on the boundary integral equations method, which can solve linear elliptic partial differential equations on arbitrary 2D domains. The models presented in this paper do not require retraining when solving partial differential equations defined on new 2D geometric domains after initial training. By introducing boundary parameter equations, we transform the problem of solving 2D partial differential equations on different geometric domains into solving 1D boundary integral equations defined on a fixed interval. This approach ensures that the models do not require retraining when solving partial differential equations on new solution domains. Moreover, due to the dimensionality reduction property of boundary integral equations, generating training data for neural operators learning boundary integral equations is easier than for neural operators learning partial differential equations. We further demonstrate the application of these models to address potential flow problems, elastostatic problems, and acoustic obstacle scattering problems. These applications demonstrate the models' effectiveness in addressing partial differential equations in different 2D domains.