UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs

Xi Han, Fei Hou, Hong Qin
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:17354-17373, 2024.

Abstract

Numerical solvers of Partial Differential Equations (PDEs) are of fundamental significance to science and engineering. To date, the historical reliance on legacy techniques has circumscribed possible integration of big data knowledge and exhibits sub-optimal efficiency for certain PDE formulations, while data-driven neural methods typically lack mathematical guarantee of convergence and correctness. This paper articulates a mathematically rigorous neural solver for linear PDEs. The proposed UGrid solver, built upon the principled integration of U-Net and MultiGrid, manifests a mathematically rigorous proof of both convergence and correctness, and showcases high numerical accuracy, as well as strong generalization power to various input geometry/values and multiple PDE formulations. In addition, we devise a new residual loss metric, which enables unsupervised training and affords more stability and a larger solution space over the legacy losses.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-han24a, title = {{UG}rid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear {PDE}s}, author = {Han, Xi and Hou, Fei and Qin, Hong}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {17354--17373}, 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/han24a/han24a.pdf}, url = {https://proceedings.mlr.press/v235/han24a.html}, abstract = {Numerical solvers of Partial Differential Equations (PDEs) are of fundamental significance to science and engineering. To date, the historical reliance on legacy techniques has circumscribed possible integration of big data knowledge and exhibits sub-optimal efficiency for certain PDE formulations, while data-driven neural methods typically lack mathematical guarantee of convergence and correctness. This paper articulates a mathematically rigorous neural solver for linear PDEs. The proposed UGrid solver, built upon the principled integration of U-Net and MultiGrid, manifests a mathematically rigorous proof of both convergence and correctness, and showcases high numerical accuracy, as well as strong generalization power to various input geometry/values and multiple PDE formulations. In addition, we devise a new residual loss metric, which enables unsupervised training and affords more stability and a larger solution space over the legacy losses.} }
Endnote
%0 Conference Paper %T UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs %A Xi Han %A Fei Hou %A Hong Qin %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-han24a %I PMLR %P 17354--17373 %U https://proceedings.mlr.press/v235/han24a.html %V 235 %X Numerical solvers of Partial Differential Equations (PDEs) are of fundamental significance to science and engineering. To date, the historical reliance on legacy techniques has circumscribed possible integration of big data knowledge and exhibits sub-optimal efficiency for certain PDE formulations, while data-driven neural methods typically lack mathematical guarantee of convergence and correctness. This paper articulates a mathematically rigorous neural solver for linear PDEs. The proposed UGrid solver, built upon the principled integration of U-Net and MultiGrid, manifests a mathematically rigorous proof of both convergence and correctness, and showcases high numerical accuracy, as well as strong generalization power to various input geometry/values and multiple PDE formulations. In addition, we devise a new residual loss metric, which enables unsupervised training and affords more stability and a larger solution space over the legacy losses.
APA
Han, X., Hou, F. & Qin, H.. (2024). UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:17354-17373 Available from https://proceedings.mlr.press/v235/han24a.html.

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