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Title: Learning domain-independent Green’s function for elliptic partial differential equations
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.  more » « less
Award ID(s):
2309798 1927432
PAR ID:
10518795
Author(s) / Creator(s):
; ; ; ;
Editor(s):
Lorenzis, Laura
Publisher / Repository:
Elsevier
Date Published:
Journal Name:
Computer Methods in Applied Mechanics and Engineering
Volume:
421
Issue:
C
ISSN:
0045-7825
Page Range / eLocation ID:
116779
Subject(s) / Keyword(s):
Boundary integral method Green's function Domain-independent Kernel methods
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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