mlsanityData to 7 Oct 2026

Multi-Scale Message Passing Neural PDE Solvers

arXiv 2302.03580 ↗First posted 7 Feb 2023cs.LG cs.NA math.NAcited by 12

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Abstract (arXiv)

We propose a novel multi-scale message passing neural network algorithm for learning the solutions of time-dependent PDEs. Our algorithm possesses both temporal and spatial multi-scale resolution features by incorporating multi-scale sequence models and graph gating modules in the encoder and processor, respectively. Benchmark numerical experiments are presented to demonstrate that the proposed algorithm outperforms baselines, particularly on a PDE with a range of spatial and temporal scales.

Key facts. Cited by 12 papers in the map (data through 7 Oct 2026). Builds on 13 of its 18 references that are in the map. First posted on arXiv 7 Feb 2023.

Builds on

13 of 18 references.

References · 13

  1. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  4. 2018Relational inductive biases, deep learning, and graph networkscited by 1,341
  5. 2022Message Passing Neural PDE Solverscited by 368
  6. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  7. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  8. 2021Learned Coarse Models for Efficient Turbulence Simulationcited by 90
  9. 2022Gradient Gating for Deep Multi-Rate Learning on Graphscited by 49
  10. 2021Long Expressive Memory for Sequence Modelingcited by 41
  11. 2021Simulating Continuum Mechanics with Multi-Scale Graph Neural Networkscited by 39
  12. 2019Deep learning observables in computational fluid dynamicscited by 38
1 more
  1. 2022MAgNet: Mesh Agnostic Neural PDE Solvercited by 28

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 6; 2024: 3; 2025: 2; 2026: 1.

12 papers.

  1. 2023How does over-squashing affect the power of GNNs?cited by 53
  2. 2023Self-Supervised Learning with Lie Symmetries for Partial Differential Equationscited by 29
  3. 2023Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domainscited by 20
  4. 2023Improving day-ahead Solar Irradiance Time Series Forecasting by Leveraging Spatio-Temporal Contextcited by 13
  5. 2024NeurIPS 2024 ML4CFD Competition: Harnessing Machine Learning for Computational Fluid Dynamics in Airfoil Designcited by 8
  6. 2025A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEscited by 3
  7. 2023DON-LSTM: Multi-Resolution Learning with DeepONets and Long Short-Term Memory Neural Networkscited by 1
  8. 2024Physics-enhanced Neural Operator for Simulating Turbulent Transportcited by 1
  9. 2023Reconstructing Turbulent Flows Using Physics-Aware Spatio-Temporal Dynamics and Test-Time Refinementcited by 0
  10. 2024Modeling Continuous Spatial-temporal Dynamics of Turbulent Flow with Test-time Refinementcited by 0
  11. 2025NeurIPS 2024 ML4CFD Competition: Results and Retrospective Analysiscited by 0
  12. 2026Equivariant Covariance Tensors: Guaranteed SPD Uncertainty for Tensor-Valued Geometric Learningcited by 0