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Neural Green’s function for Laplacian systems

Authors :: Tang, Jingwei
Azevedo, Vinicius C.
Cordonnier, Guillaume
Solenthaler, Barbara
Computer Graphics Laboratory [ETH Zurich]
Disney Research Zürich (DRZ)
GRAPHics and DEsign with hEterogeneous COntent (GRAPHDECO)
Inria Sophia Antipolis - Méditerranée (CRISAM)
Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Source :: Computers and Graphics, Computers and Graphics, 2022, 107, pp.186-196. ⟨10.1016/j.cag.2022.07.016⟩, Computers & Graphics, 107
Publication Year :: 2022
Publisher :: Elsevier BV, 2022.
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.<br />Computers & Graphics, 107<br />ISSN:0097-8493<br />ISSN:1873-7684