# Multi-Scale Message Passing Neural PDE Solvers

arXiv: https://arxiv.org/abs/2302.03580 · HTML: https://mlsanity.com/paper/2302.03580/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 18 references that are in the map.
- First posted on arXiv 7 Feb 2023.

## Builds on (most cited first, 13 of 13)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [Gradient Gating for Deep Multi-Rate Learning on Graphs](https://mlsanity.com/paper/2210.00513/) (2022)
- [Long Expressive Memory for Sequence Modeling](https://mlsanity.com/paper/2110.04744/) (2021)
- [Simulating Continuum Mechanics with Multi-Scale Graph Neural Networks](https://mlsanity.com/paper/2106.04900/) (2021)
- [Deep learning observables in computational fluid dynamics](https://mlsanity.com/paper/1903.03040/) (2019)
- [MAgNet: Mesh Agnostic Neural PDE Solver](https://mlsanity.com/paper/2210.05495/) (2022)

## Built on by (most cited first, 12 of 12)

- [How does over-squashing affect the power of GNNs?](https://mlsanity.com/paper/2306.03589/) (2023)
- [Self-Supervised Learning with Lie Symmetries for Partial Differential Equations](https://mlsanity.com/paper/2307.05432/) (2023)
- [Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains](https://mlsanity.com/paper/2305.19663/) (2023)
- [Improving day-ahead Solar Irradiance Time Series Forecasting by Leveraging Spatio-Temporal Context](https://mlsanity.com/paper/2306.01112/) (2023)
- [NeurIPS 2024 ML4CFD Competition: Harnessing Machine Learning for Computational Fluid Dynamics in Airfoil Design](https://mlsanity.com/paper/2407.01641/) (2024)
- [A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEs](https://mlsanity.com/paper/2502.08683/) (2025)
- [DON-LSTM: Multi-Resolution Learning with DeepONets and Long Short-Term Memory Neural Networks](https://mlsanity.com/paper/2310.02491/) (2023)
- [Physics-enhanced Neural Operator for Simulating Turbulent Transport](https://mlsanity.com/paper/2406.04367/) (2024)
- [NeurIPS 2024 ML4CFD Competition: Results and Retrospective Analysis](https://mlsanity.com/paper/2506.08516/) (2025)
- [Reconstructing Turbulent Flows Using Physics-Aware Spatio-Temporal Dynamics and Test-Time Refinement](https://mlsanity.com/paper/2304.12130/) (2023)
- [Modeling Continuous Spatial-temporal Dynamics of Turbulent Flow with Test-time Refinement](https://mlsanity.com/paper/2412.19927/) (2024)
- [Equivariant Covariance Tensors: Guaranteed SPD Uncertainty for Tensor-Valued Geometric Learning](https://mlsanity.com/paper/2608.24386/) (2026)

## 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.