filename : Tan22a.pdf entry : article conference : pages : 186-196 year : 2022 month : October title : Neural Green’s function for Laplacian systems subtitle : author : Jingwei Tang, Vinicius C. Azevedo, Guillaume Cordonnier, Barbara Solenthaler booktitle : Computers & Graphics ISSN/ISBN : 0097-8493 editor : publisher : Elsevier publ.place : volume : 107 issue : language : English keywords : abstract : Solving linear system of equations stemming from Laplacian operators is at the heart of a wide range of applications. Due to the sparsity of the linear systems, iterative solvers such as Conjugate Gradient and Multigrid are usually employed when the solution has a large number of degrees of freedom. These iterative solvers can be seen as sparse approximations of the Green’s function for the Laplacian operator. In this paper we propose a machine learning approach that regresses a Green’s function from boundary conditions. This is enabled by a Green’s function that can be effectively represented in a multi-scale fashion, drastically reducing the cost associated with a dense matrix representation. Additionally, since the Green’s function is solely dependent on boundary conditions, training the proposed neural network does not require sampling the right-hand side of the linear system. We show results that our method outperforms state of the art Conjugate Gradient and Multigrid methods.