# Physics-Informed Coarsening for Multigrid Graph Neural Surrogates

arXiv: https://arxiv.org/abs/2605.31013 · HTML: https://mlsanity.com/paper/2605.31013/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [MultiScale MeshGraphNets](https://mlsanity.com/paper/2210.00612/) (2022)
- [Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries](https://mlsanity.com/paper/2502.02414/) (2025)
- [Predicting Physics in Mesh-reduced Space with Temporal Attention](https://mlsanity.com/paper/2201.09113/) (2022)
- [Latent Neural Operator for Solving Forward and Inverse PDE Problems](https://mlsanity.com/paper/2406.03923/) (2024)
- [Deep convolutional encoder-decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media](https://mlsanity.com/paper/1807.00882/) (2018)
- [Simulating Continuum Mechanics with Multi-Scale Graph Neural Networks](https://mlsanity.com/paper/2106.04900/) (2021)
- [Learning Flexible Body Collision Dynamics with Hierarchical Contact Mesh Transformer](https://mlsanity.com/paper/2312.12467/) (2023)
- [U-net architectures for fast prediction of incompressible laminar flows](https://mlsanity.com/paper/1910.13532/) (2019)
- [MG-GNN: Multigrid Graph Neural Networks for Learning Multilevel Domain Decomposition Methods](https://mlsanity.com/paper/2301.11378/) (2023)
- [GNN-based physics solver for time-independent PDEs](https://mlsanity.com/paper/2303.15681/) (2023)
- [Multi-Grid Graph Neural Networks with Self-Attention for Computational Mechanics](https://mlsanity.com/paper/2409.11899/) (2024)
- [Training Transformers for Mesh-Based Simulations](https://mlsanity.com/paper/2508.18051/) (2025)
- [Up-sampling-only and Adaptive Mesh-based GNN for Simulating Physical Systems](https://mlsanity.com/paper/2409.04740/) (2024)
- [Graph Network-based Structural Simulator: Graph Neural Networks for Structural Dynamics](https://mlsanity.com/paper/2510.25683/) (2025)

## Abstract (arXiv)

Learning-based surrogates for partial differential equations have recently matched the accuracy of classical solvers while achieving orders-of-magnitude speedups, predominantly in fluid settings and structured geometries. In contrast, robust surrogates for deformable solids remain underexplored, despite the presence of nonlinear elasticity, plasticity, and transient behavior that challenge standard architectures. We introduce a multigrid graph neural network for solid mechanics that couples an encoder-processor-decoder backbone with a physics-informed coarsening strategy. Instead of downsampling via geometric heuristics, our method scores nodes using a residual-based measure of local physical activity and preferentially retains regions of high strain or stress concentration, allocating multiscale capacity where it is most needed. This preserves long-range interactions through hierarchical message passing while improving stability over long rollouts. We evaluate on multiple datasets covering linear, nonlinear, and transient regimes, and observe consistent gains in accuracy and rollout stability compared to standard sampling baselines. Our results highlight the importance of physics-informed coarsening for scalable surrogate modeling in solid mechanics.