mlsanityData to 7 Oct 2026

Message Passing Neural PDE Solvers

arXiv 2202.03376 ↗First posted 7 Feb 2022cs.LG cs.CV cs.NAcited by 368

Not on the map. Why?

Cited by 25 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.

Key facts. Cited by 368 papers in the map (data through 7 Oct 2026). Builds on 27 of its 52 references that are in the map. First posted on arXiv 7 Feb 2022.

Builds on

27 of 52 references.

References · 27

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,282
  3. 2015Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shiftcited by 6,979
  4. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  5. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  6. 2016Convolutional Neural Networks on Graphs with Fast Localized Spectral Filteringcited by 2,085
  7. 2018Relational inductive biases, deep learning, and graph networkscited by 1,341
  8. 2016Instance Normalization: The Missing Ingredient for Fast Stylizationcited by 1,187
  9. 2015Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)cited by 1,061
  10. 2021E(n) Equivariant Graph Neural Networkscited by 951
  11. 2020Learning to Simulate Complex Physics with Graph Networkscited by 867
  12. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
15 more
  1. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  2. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  3. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  4. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
  5. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  6. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  7. 2019Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networkscited by 144
  8. 2020Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Predictioncited by 137
  9. 2019Learning to Optimize Multigrid PDE Solverscited by 79
  10. 2019Learning Neural PDE Solvers with Convergence Guaranteescited by 77
  11. 2020A physics-informed operator regression framework for extracting data-driven continuum modelscited by 60
  12. 2015Probabilistic Numerics and Uncertainty in Computationscited by 59
  13. 2019Combining Generative and Discriminative Models for Hybrid Inferencecited by 34
  14. 2021Boundary Graph Neural Networks for 3D Simulationscited by 24
  15. 2014Probabilistic ODE Solvers with Runge-Kutta Meanscited by 22

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 35; 2023: 69; 2024: 75; 2025: 83; 2026: 106.

368 papers.

Sort:

By area:

  1. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 396
  2. 2023ClimaX: A foundation model for weather and climatecited by 325
  3. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
  4. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 205
  5. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
  6. 2024Poseidon: Efficient Foundation Models for PDEscited by 186
  7. 2022Forecasting Global Weather with Graph Neural Networkscited by 165
  8. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstructioncited by 150
  9. 2024DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Trainingcited by 137
  10. 2023Scalable Transformer for PDE Surrogate Modelingcited by 123
  11. 2023DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecastingcited by 105
  12. 2022Clifford Neural Layers for PDE Modelingcited by 91
38 more
  1. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  2. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  3. 2023Operator Learning with Neural Fields: Tackling PDEs on General Geometriescited by 80
  4. 2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 76
  5. 2022Continuous PDE Dynamics Forecasting with Implicit Neural Representationscited by 71
  6. 2022A Survey on Graph Representation Learning Methodscited by 66
  7. 2023Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulationcited by 65
  8. 2023Neural Inverse Operators for Solving PDE Inverse Problemscited by 61
  9. 2023How does over-squashing affect the power of GNNs?cited by 53
  10. 2023From Zero to Turbulence: Generative Modeling for 3D Flow Simulationcited by 46
  11. 2022Mitigating spectral bias for the multiscale operator learningcited by 43
  12. 2023Training neural operators to preserve invariant measures of chaotic attractorscited by 43
  13. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 42
  14. 2023MgNO: Efficient Parameterization of Linear Operators via Multigridcited by 40
  15. 2022Learning to Accelerate Partial Differential Equations via Latent Global Evolutioncited by 38
  16. 2024AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fieldscited by 38
  17. 2023Geometric Clifford Algebra Networkscited by 37
  18. 2023Learning Neural PDE Solvers with Parameter-Guided Channel Attentioncited by 37
  19. 2024Equivariant Graph Neural Operator for Modeling 3D Dynamicscited by 37
  20. 2025CodePDE: An Inference Framework for LLM-driven PDE Solver Generationcited by 37
  21. 2022Koopman Neural Forecaster for Time Series with Temporal Distribution Shiftscited by 35
  22. 2022Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditionscited by 33
  23. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 33
  24. 2022ACMP: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networkscited by 31
  25. 2024Unisolver: PDE-Conditional Transformers Towards Universal Neural PDE Solverscited by 31
  26. 2022Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamicscited by 30
  27. 2024PDE Generalization of In-Context Operator Networks: A Study on 1D Scalar Nonlinear Conservation Lawscited by 30
  28. 2024DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systemscited by 29
  29. 2022MAgNet: Mesh Agnostic Neural PDE Solvercited by 28
  30. 2022INO: Invariant Neural Operators for Learning Complex Physical Systems with Momentum Conservationcited by 28
  31. 2024Wavelet Diffusion Neural Operatorcited by 28
  32. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  33. 2023LagrangeBench: A Lagrangian Fluid Mechanics Benchmarking Suitecited by 26
  34. 2024Universal Physics Transformers: A Framework For Efficiently Scaling Neural Operatorscited by 25
  35. 2023Lie Point Symmetry and Physics Informed Networkscited by 24
  36. 2024HAMLET: Graph Transformer Neural Operator for Partial Differential Equationscited by 24
  37. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 24
  38. 2023A Neural PDE Solver with Temporal Stencil Modelingcited by 22

50 most cited of 368 All 368 (JSON)