Neural Green?s function for Laplacian systems?

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. (c) 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

This paper proposes a machine learning approach to regress a Green's function from boundary conditions for solving linear systems of equations stemming from Laplacian operators. The proposed method effectively represents the Green's function in a multi-scale fashion, reducing the cost associated with a dense matrix representation. Moreover, training the neural network does not require sampling the right-hand side of the linear system. Experimental results demonstrate that the proposed method outperforms state-of-the-art Conjugate Gradient and Multigrid methods.