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.

@Article{TACS22,
  author       = "Tang, Jingwei and Azevedo, Vinicius and Cordonnier, Guillaume and Solenthaler, Barbara",
  title        = "Neural Green’s Function for Laplacian Systems",
  journal      = "Computers \& Graphics",
  volume       = "107",
  pages        = "186--196",
  year         = "2022",
  keywords     = "Machine learning, Modeling and simulation, Poisson equation, Green’s function",
  url          = "http://www-sop.inria.fr/reves/Basilic/2022/TACS22"
}