UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs
Published in Proceedings of the 41st International Conference on Machine Learning (ICML), 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 self-supervised training and affords more stability and a larger solution space over the legacy losses.
More information are available at arXiv, openreview, GitHub, and ICML 2024 Virtual Poster Session.
Recommended citation: Xi Han, Fei Hou and Hong Qin, "UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs", In Proceedings of the 41st International Conference on Machine Learning (ICML), 2024.
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