mlsanityData to 7 Oct 2026

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

arXiv 2408.04846 ↗First posted 9 Aug 2024math.NA cs.AI cs.LGcited by 6

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Abstract (arXiv)

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.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 15 of its 33 references that are in the map. First posted on arXiv 9 Aug 2024.

Builds on

15 of 33 references.

References · 15

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  4. 2016Accelerating Eulerian Fluid Simulation With Convolutional Networkscited by 128
  5. 2019Learning to Optimize Multigrid PDE Solverscited by 79
  6. 2019Learning Neural PDE Solvers with Convergence Guaranteescited by 77
  7. 2020Learning Algebraic Multigrid Using Graph Neural Networkscited by 54
  8. 2021Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimalitycited by 40
  9. 2020nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applicationscited by 36
  10. 2017Deep Learning the Physics of Transport Phenomenacited by 26
  11. 2019Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian meshcited by 22
  12. 2021Parametric Complexity Bounds for Approximating PDEs with Neural Networkscited by 18
3 more
  1. 2018Weakly-Supervised Deep Learning of Heat Transport via Physics Informed Losscited by 15
  2. 2018Solving Poisson's Equation using Deep Learning in Particle Simulation of PN Junctioncited by 4
  3. 2018AMGCL: an Efficient, Flexible, and Extensible Algebraic Multigrid Implementationcited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 3; 2026: 3.

6 papers.

  1. 2025NOWS: Neural Operator Warm Starts for Accelerating Iterative Solverscited by 9
  2. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  3. 2025Universal Fourier Neural Operators for periodic homogenization problems in linear elasticitycited by 3
  4. 2025GeoMaNO: Geometric Mamba Neural Operator for Partial Differential Equationscited by 2
  5. 2026Automated Grammar-based Algebraic Multigrid Design With Evolutionary Algorithmscited by 0
  6. 2026RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Correctionscited by 0