BI-GreenNet: Learning Green’s Functions by Boundary Integral Network

Abstract

Green’s function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, in most cases, Green’s function is difficult to compute. The troubles arise in the following threefold. Firstly, compared with the original PDE, the dimension of Green’s function is doubled, making it impossible to be handled by traditional mesh-based methods. Secondly, Green’s function usually contains singularities which increase in the difficulty to get a good approximation. Lastly, the computational domain may be very complex or even unbounded. To override these problems, we develop a new framework for computing Green’s function leveraging the fundamental solution, boundary integral method and neural networks with reasonably high accuracy in this paper. We focus on Green’s function of Poisson and Helmholtz equations in bounded domains, unbounded domains. We also consider Poisson equation and Helmholtz equations in domains with interfaces. Extensive numerical experiments illustrate the efficiency and accuracy of our method for solving Green’s function. In addition, Green’s function provides the operator from the source term and boundary condition to the PDE solution. We apply Green’s function to solve PDEs with different sources, and obtain reasonably high-precision solutions, which shows the good generalization ability of our method. However, the requirements for explicit fundamental solutions to remove the singularity of Green’s function hinder the application of our method in more complex PDEs, such as variable coefficient equations, which will be investigated in our future work.