mlsanityData to 14 Oct 2026

On the under-reaching phenomenon in message-passing neural PDE solvers: revisiting the CFL condition

arXiv 2507.08861 ↗First posted 9 Jul 2025cs.LG stat.MLcited by 4

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Abstract (arXiv)

This paper proposes sharp lower bounds for the number of message passing iterations required in graph neural networks (GNNs) when solving partial differential equations (PDE). This significantly reduces the need for exhaustive hyperparameter tuning. Bounds are derived for the three fundamental classes of PDEs (hyperbolic, parabolic and elliptic) by relating the physical characteristics of the problem in question to the message-passing requirement of GNNs. In particular, we investigate the relationship between the physical constants of the equations governing the problem, the spatial and temporal discretisation and the message passing mechanisms in GNNs. When the number of message passing iterations is below these proposed limits, information does not propagate efficiently through the network, resulting in poor solutions, even for deep GNN architectures. In contrast, when the suggested lower bound is satisfied, the GNN parameterisation allows the model to accurately capture the underlying phenomenology, resulting in solvers of adequate accuracy. Examples are provided for four different examples of equations that show the sharpness of the proposed lower bounds.

Key facts. Cited by 4 papers in the map (data through 14 Oct 2026). Builds on 28 of its 44 references that are in the map. First posted on arXiv 9 Jul 2025.

Builds on

28 of 44 references.

References · 28

  1. 2019A Comprehensive Survey on Graph Neural Networkscited by 2,919
  2. 2016Geometric deep learning: going beyond Euclidean datacited by 1,048
  3. 2020Learning Mesh-Based Simulation with Graph Networkscited by 868
  4. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  5. 2021Understanding over-squashing and bottlenecks on graphs via curvaturecited by 478
  6. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  7. 2022Message Passing Neural PDE Solverscited by 386
  8. 2022GraphCast: Learning skillful medium-range global weather forecastingcited by 267
  9. 2020A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEscited by 105
  10. 2021Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problemscited by 102
  11. 2021Breaking the Limits of Message Passing Graph Neural Networkscited by 101
  12. 2021POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decompositioncited by 79
16 more
  1. 2020Structure-preserving neural networkscited by 55
  2. 2020DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolationcited by 45
  3. 2022Learning rigid dynamics with face interaction graph networkscited by 43
  4. 2022Thermodynamics-informed graph neural networkscited by 33
  5. 2023Reversible and irreversible bracket-based dynamics for deep graph neural networkscited by 26
  6. 2022Unravelling the Performance of Physics-informed Graph Neural Networks for Dynamical Systemscited by 23
  7. 2024A review of graph neural network applications in mechanics-related domainscited by 14
  8. 2023NodeMixup: Tackling Under-Reaching for Graph Neural Networkscited by 12
  9. 2022FlowGNN: A Dataflow Architecture for Real-Time Workload-Agnostic Graph Neural Network Inferencecited by 11
  10. 2022Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systemscited by 11
  11. 2022A Thermodynamics-informed Active Learning Approach to Perception and Reasoning about Fluidscited by 9
  12. 2022Thermodynamics-informed neural networks for physically realistic mixed realitycited by 9
  13. 2022Physics-Informed Graph Neural Network for Spatial-temporal Production Forecastingcited by 8
  14. 2024On the feasibility of foundational models for the simulation of physical phenomenacited by 7
  15. 2024Handling geometrical variability in nonlinear reduced order modeling through Continuous Geometry-Aware DL-ROMscited by 4
  16. 2024Thermodynamics-informed super-resolution of scarce temporal dynamics datacited by 3

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 4.

4 papers.

  1. 2026MeshGraphNet-Transformer: Scalable Mesh-based Learned Simulation for Solid Mechanicscited by 10
  2. 2026Variational Graph Neural Networks for Uncertainty Quantification in Inverse Problemscited by 0
  3. 2026Towards Fast GNN Surrogates for CO2 Migration in Complex Geological Formationscited by 0
  4. 2026Attention mechanism for scalable mesh-based neural surrogates of free-surface fluidscited by 0