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

arXiv: https://arxiv.org/abs/2507.08861 · HTML: https://mlsanity.com/paper/2507.08861/

## 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 (most cited first, 28 of 28)

- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [GraphCast: Learning skillful medium-range global weather forecasting](https://mlsanity.com/paper/2212.12794/) (2022)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Breaking the Limits of Message Passing Graph Neural Networks](https://mlsanity.com/paper/2106.04319/) (2021)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [Structure-preserving neural networks](https://mlsanity.com/paper/2004.04653/) (2020)
- [DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation](https://mlsanity.com/paper/2012.02681/) (2020)
- [Learning rigid dynamics with face interaction graph networks](https://mlsanity.com/paper/2212.03574/) (2022)
- [Thermodynamics-informed graph neural networks](https://mlsanity.com/paper/2203.01874/) (2022)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [Unravelling the Performance of Physics-informed Graph Neural Networks for Dynamical Systems](https://mlsanity.com/paper/2211.05520/) (2022)
- [A review of graph neural network applications in mechanics-related domains](https://mlsanity.com/paper/2407.11060/) (2024)
- [NodeMixup: Tackling Under-Reaching for Graph Neural Networks](https://mlsanity.com/paper/2312.13032/) (2023)
- [FlowGNN: A Dataflow Architecture for Real-Time Workload-Agnostic Graph Neural Network Inference](https://mlsanity.com/paper/2204.13103/) (2022)
- [Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systems](https://mlsanity.com/paper/2211.01873/) (2022)
- [A Thermodynamics-informed Active Learning Approach to Perception and Reasoning about Fluids](https://mlsanity.com/paper/2203.05775/) (2022)
- [Thermodynamics-informed neural networks for physically realistic mixed reality](https://mlsanity.com/paper/2210.13414/) (2022)
- [Physics-Informed Graph Neural Network for Spatial-temporal Production Forecasting](https://mlsanity.com/paper/2209.11885/) (2022)
- [On the feasibility of foundational models for the simulation of physical phenomena](https://mlsanity.com/paper/2410.14645/) (2024)
- [Handling geometrical variability in nonlinear reduced order modeling through Continuous Geometry-Aware DL-ROMs](https://mlsanity.com/paper/2411.05486/) (2024)
- [Thermodynamics-informed super-resolution of scarce temporal dynamics data](https://mlsanity.com/paper/2402.17506/) (2024)

## Built on by (most cited first, 4 of 4)

- [MeshGraphNet-Transformer: Scalable Mesh-based Learned Simulation for Solid Mechanics](https://mlsanity.com/paper/2601.23177/) (2026)
- [Variational Graph Neural Networks for Uncertainty Quantification in Inverse Problems](https://mlsanity.com/paper/2603.29515/) (2026)
- [Towards Fast GNN Surrogates for CO2 Migration in Complex Geological Formations](https://mlsanity.com/paper/2606.17180/) (2026)
- [Attention mechanism for scalable mesh-based neural surrogates of free-surface fluids](https://mlsanity.com/paper/2606.23251/) (2026)

## 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.