Neural Green's function for Laplacian systems
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Date
2022-10
Publication Type
Journal Article
ETH Bibliography
yes
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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.
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Publication status
published
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Book title
Journal / series
Volume
107
Pages / Article No.
186 - 196
Publisher
Elsevier
Event
Edition / version
Methods
Software
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Date collected
Date created
Subject
Machine learning; Modeling and simulation; Poisson equation; Green’s function
Organisational unit
03420 - Gross, Markus / Gross, Markus
Notes
Funding
ETH-08 18-1 - A technical foundation for deep learning based physics simulations (ETHZ)