Work

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

Year: 2024

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2408.04846

Authors: Han, Xi, Hou, Fei, Qin, Hong

Institutions: Chinese Academy of Sciences, Institute of Software, State Key Laboratory of Computer Science, Stony Brook University, University of Chinese Academy of Sciences

Language: English

Topic: Model Reduction and Neural Networks

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: CHS: Small: Novel Data-adaptive Analytics for Manifold Informatics: Theory, Algorithms, and Applications, CHS: Small: Novel Method for Vectorization of Arbitrary Natural Images and Its Applications

OpenAlex ID: W6966960820

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