Learning Domain-Independent Green's Function For Elliptic Partial
  Differential Equations
Pawan Negi Maggie Cheng Mahesh Krishnamurthy Wenjun Ying Shuwang Li
Green's function characterizes a partial differential equation (PDE) and maps
its solution in the entire domain as integrals. Finding the analytical form of
Green's function is a non-trivial exercise, especially for a PDE defined on a
complex domain or a PDE with variable coefficients. In this paper, we propose a
novel boundary integral network to learn the domain-independent Green's
function, referred to as BIN-G. We evaluate the Green's function in the BIN-G
using a radial basis function (RBF) kernel-based neural network. We train the
BIN-G by minimizing the residual of the PDE and the mean squared errors of the
solutions to the boundary integral equations for prescribed test functions. By
leveraging the symmetry of the Green's function and controlling refinements of
the RBF kernel near the singularity of the Green function, we demonstrate that
our numerical scheme enables fast training and accurate evaluation of the
Green's function for PDEs with variable coefficients. The learned Green's
function is independent of the domain geometries, forcing terms, and boundary
conditions in the boundary integral formulation. Numerical experiments verify
the desired properties of the method and the expected accuracy for the
two-dimensional Poisson and Helmholtz equations with variable coefficients.