Solving Poisson Equations using Neural Walk-on-Spheres

Hong Chul Nam, Julius Berner, Anima Anandkumar
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:37277-37292, 2024.

Abstract

We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. Leveraging stochastic representations and Walk-on-Spheres methods, we develop novel losses for neural networks based on the recursive solution of Poisson equations on spheres inside the domain. The resulting method is highly parallelizable and does not require spatial gradients for the loss. We provide a comprehensive comparison against competing methods based on PINNs, the Deep Ritz method, and (backward) stochastic differential equations. In several challenging, high-dimensional numerical examples, we demonstrate the superiority of NWoS in accuracy, speed, and computational costs. Compared to commonly used PINNs, our approach can reduce memory usage and errors by orders of magnitude. Furthermore, we apply NWoS to problems in PDE-constrained optimization and molecular dynamics to show its efficiency in practical applications.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-nam24a, title = {Solving Poisson Equations using Neural Walk-on-Spheres}, author = {Nam, Hong Chul and Berner, Julius and Anandkumar, Anima}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {37277--37292}, 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/nam24a/nam24a.pdf}, url = {https://proceedings.mlr.press/v235/nam24a.html}, abstract = {We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. Leveraging stochastic representations and Walk-on-Spheres methods, we develop novel losses for neural networks based on the recursive solution of Poisson equations on spheres inside the domain. The resulting method is highly parallelizable and does not require spatial gradients for the loss. We provide a comprehensive comparison against competing methods based on PINNs, the Deep Ritz method, and (backward) stochastic differential equations. In several challenging, high-dimensional numerical examples, we demonstrate the superiority of NWoS in accuracy, speed, and computational costs. Compared to commonly used PINNs, our approach can reduce memory usage and errors by orders of magnitude. Furthermore, we apply NWoS to problems in PDE-constrained optimization and molecular dynamics to show its efficiency in practical applications.} }
Endnote
%0 Conference Paper %T Solving Poisson Equations using Neural Walk-on-Spheres %A Hong Chul Nam %A Julius Berner %A Anima Anandkumar %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-nam24a %I PMLR %P 37277--37292 %U https://proceedings.mlr.press/v235/nam24a.html %V 235 %X We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. Leveraging stochastic representations and Walk-on-Spheres methods, we develop novel losses for neural networks based on the recursive solution of Poisson equations on spheres inside the domain. The resulting method is highly parallelizable and does not require spatial gradients for the loss. We provide a comprehensive comparison against competing methods based on PINNs, the Deep Ritz method, and (backward) stochastic differential equations. In several challenging, high-dimensional numerical examples, we demonstrate the superiority of NWoS in accuracy, speed, and computational costs. Compared to commonly used PINNs, our approach can reduce memory usage and errors by orders of magnitude. Furthermore, we apply NWoS to problems in PDE-constrained optimization and molecular dynamics to show its efficiency in practical applications.
APA
Nam, H.C., Berner, J. & Anandkumar, A.. (2024). Solving Poisson Equations using Neural Walk-on-Spheres. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:37277-37292 Available from https://proceedings.mlr.press/v235/nam24a.html.

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