A Neural-Preconditioned Poisson Solver for Mixed Dirichlet and Neumann Boundary Conditions

Kai Weixian Lan, Elias Gueidon, Ayano Kaneda, Julian Panetta, Joseph Teran
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:25976-25994, 2024.

Abstract

We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. Typical Poisson discretizations yield large, ill-conditioned linear systems. Iterative solvers can be effective for these problems, but only when equipped with powerful preconditioners. Unfortunately, effective preconditioners like multigrid require costly setup phases that must be re-executed every time domain shapes or boundary conditions change, forming a severe bottleneck for problems with evolving boundaries. In contrast, we present a neural preconditioner trained to efficiently approximate the inverse of the discrete Laplacian in the presence of such changes. Our approach generalizes to domain shapes, boundary conditions, and grid sizes outside the training set. The key to our preconditioner’s success is a novel, lightweight neural network architecture featuring spatially varying convolution kernels and supporting fast inference. We demonstrate that our solver outperforms state-of-the-art methods like algebraic multigrid as well as recently proposed neural preconditioners on challenging test cases arising from incompressible fluid simulations.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-lan24a, title = {A Neural-Preconditioned Poisson Solver for Mixed {D}irichlet and Neumann Boundary Conditions}, author = {Lan, Kai Weixian and Gueidon, Elias and Kaneda, Ayano and Panetta, Julian and Teran, Joseph}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {25976--25994}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/lan24a/lan24a.pdf}, url = {https://proceedings.mlr.press/v235/lan24a.html}, abstract = {We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. Typical Poisson discretizations yield large, ill-conditioned linear systems. Iterative solvers can be effective for these problems, but only when equipped with powerful preconditioners. Unfortunately, effective preconditioners like multigrid require costly setup phases that must be re-executed every time domain shapes or boundary conditions change, forming a severe bottleneck for problems with evolving boundaries. In contrast, we present a neural preconditioner trained to efficiently approximate the inverse of the discrete Laplacian in the presence of such changes. Our approach generalizes to domain shapes, boundary conditions, and grid sizes outside the training set. The key to our preconditioner’s success is a novel, lightweight neural network architecture featuring spatially varying convolution kernels and supporting fast inference. We demonstrate that our solver outperforms state-of-the-art methods like algebraic multigrid as well as recently proposed neural preconditioners on challenging test cases arising from incompressible fluid simulations.} }
Endnote
%0 Conference Paper %T A Neural-Preconditioned Poisson Solver for Mixed Dirichlet and Neumann Boundary Conditions %A Kai Weixian Lan %A Elias Gueidon %A Ayano Kaneda %A Julian Panetta %A Joseph Teran %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-lan24a %I PMLR %P 25976--25994 %U https://proceedings.mlr.press/v235/lan24a.html %V 235 %X We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. Typical Poisson discretizations yield large, ill-conditioned linear systems. Iterative solvers can be effective for these problems, but only when equipped with powerful preconditioners. Unfortunately, effective preconditioners like multigrid require costly setup phases that must be re-executed every time domain shapes or boundary conditions change, forming a severe bottleneck for problems with evolving boundaries. In contrast, we present a neural preconditioner trained to efficiently approximate the inverse of the discrete Laplacian in the presence of such changes. Our approach generalizes to domain shapes, boundary conditions, and grid sizes outside the training set. The key to our preconditioner’s success is a novel, lightweight neural network architecture featuring spatially varying convolution kernels and supporting fast inference. We demonstrate that our solver outperforms state-of-the-art methods like algebraic multigrid as well as recently proposed neural preconditioners on challenging test cases arising from incompressible fluid simulations.
APA
Lan, K.W., Gueidon, E., Kaneda, A., Panetta, J. & Teran, J.. (2024). A Neural-Preconditioned Poisson Solver for Mixed Dirichlet and Neumann Boundary Conditions. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:25976-25994 Available from https://proceedings.mlr.press/v235/lan24a.html.

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