# Efficient Learning of Mesh-Based Physical Simulation with BSMS-GNN

arXiv: https://arxiv.org/abs/2210.02573 · HTML: https://mlsanity.com/paper/2210.02573/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 52 references that are in the map.
- First posted on arXiv 5 Oct 2022.

## Builds on (most cited first, 30 of 30)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological View](https://mlsanity.com/paper/1909.03211/) (2019)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [Transformers for Modeling Physical Systems](https://mlsanity.com/paper/2010.03957/) (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)
- [MultiScale MeshGraphNets](https://mlsanity.com/paper/2210.00612/) (2022)
- [Predicting Physics in Mesh-reduced Space with Temporal Attention](https://mlsanity.com/paper/2201.09113/) (2022)
- [Hierarchical Graph Pooling with Structure Learning](https://mlsanity.com/paper/1911.05954/) (2019)
- [Rethinking pooling in graph neural networks](https://mlsanity.com/paper/2010.11418/) (2020)
- [CFDNet: a deep learning-based accelerator for fluid simulations](https://mlsanity.com/paper/2005.04485/) (2020)
- [Simulating Continuum Mechanics with Multi-Scale Graph Neural Networks](https://mlsanity.com/paper/2106.04900/) (2021)
- [Comparing recurrent and convolutional neural networks for predicting wave propagation](https://mlsanity.com/paper/2002.08981/) (2020)
- [Direct Prediction of Steady-State Flow Fields in Meshed Domain with Graph Networks](https://mlsanity.com/paper/2105.02575/) (2021)
- [Towards Fast Simulation of Environmental Fluid Mechanics with Multi-Scale Graph Neural Networks](https://mlsanity.com/paper/2205.02637/) (2022)

## Built on by (most cited first, 30 of 45)

- [SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biases](https://mlsanity.com/paper/2308.13212/) (2023)
- [VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Prediction](https://mlsanity.com/paper/2411.16063/) (2024)
- [Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks](https://mlsanity.com/paper/2504.02843/) (2025)
- [Erwin: A Tree-based Hierarchical Transformer for Large-scale Physical Systems](https://mlsanity.com/paper/2502.17019/) (2025)
- [DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic prediction](https://mlsanity.com/paper/2504.19496/) (2025)
- [MMGP: a Mesh Morphing Gaussian Process-based machine learning method for regression of physical problems under non-parameterized geometrical variability](https://mlsanity.com/paper/2305.12871/) (2023)
- [PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations](https://mlsanity.com/paper/2412.05994/) (2024)
- [Diffusion-Based Hierarchical Graph Neural Networks for Simulating Nonlinear Solid Mechanics](https://mlsanity.com/paper/2506.06045/) (2025)
- [MeshGraphNet-Transformer: Scalable Mesh-based Learned Simulation for Solid Mechanics](https://mlsanity.com/paper/2601.23177/) (2026)
- [HEIST: A Graph Foundation Model for Spatial Transcriptomics and Proteomics Data](https://mlsanity.com/paper/2506.11152/) (2025)
- [PIORF: Physics-Informed Ollivier-Ricci Flow for Long-Range Interactions in Mesh Graph Neural Networks](https://mlsanity.com/paper/2504.04052/) (2025)
- [Improving Long-Range Interactions in Graph Neural Simulators via Hamiltonian Dynamics](https://mlsanity.com/paper/2511.08185/) (2025)
- [ParticleGS: Learning Neural Gaussian Particle Dynamics from Videos for Prior-free Physical Motion Extrapolation](https://mlsanity.com/paper/2505.20270/) (2025)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Community-Invariant Graph Contrastive Learning](https://mlsanity.com/paper/2405.01350/) (2024)
- [EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions](https://mlsanity.com/paper/2410.03779/) (2024)
- [Learning Physical Simulation with Message Passing Transformer](https://mlsanity.com/paper/2406.06060/) (2024)
- [ClothTransformer: Unified Latent-Space Transformers for Scalable Cloth Simulation](https://mlsanity.com/paper/2605.27852/) (2026)
- [A Mixture of Experts Gating Network for Enhanced Surrogate Modeling in External Aerodynamics](https://mlsanity.com/paper/2508.21249/) (2025)
- [M4GN: Mesh-based Multi-segment Hierarchical Graph Network for Dynamic Simulations](https://mlsanity.com/paper/2509.10659/) (2025)
- [From Structured to Unstructured:A Comparative Analysis of Computer Vision and Graph Models in solving Mesh-based PDEs](https://mlsanity.com/paper/2406.00081/) (2024)
- [EqCollide: Equivariant and Collision-Aware Deformable Objects Neural Simulator](https://mlsanity.com/paper/2506.05797/) (2025)
- [Curriculum Learning for Mesh-based simulations](https://mlsanity.com/paper/2509.13138/) (2025)
- [Rigid Body Adversarial Attacks](https://mlsanity.com/paper/2502.05669/) (2025)
- [PLAID: A Unified Data Model for Machine Learning on Heterogeneous Physics Simulations](https://mlsanity.com/paper/2505.02974/) (2025)
- [Handle-based Mesh Deformation Guided By Vision Language Model](https://mlsanity.com/paper/2506.04562/) (2025)
- [Learning Deformable Body Interactions With Adaptive Spatial Tokenization](https://mlsanity.com/paper/2507.13707/) (2025)
- [Planning-Query-Guided Model Generation for Model-Based Deformable Object Manipulation](https://mlsanity.com/paper/2508.19199/) (2025)
- [TGLF-WINN: Data-Efficient Deep Learning Surrogate for Turbulent Transport Modeling in Fusion](https://mlsanity.com/paper/2509.07024/) (2025)
- [PIBNet: a Physics-Inspired Boundary Network for Multiple Scattering Simulations](https://mlsanity.com/paper/2512.02049/) (2025)

## Abstract (arXiv)

Learning the physical simulation on large-scale meshes with flat Graph Neural Networks (GNNs) and stacking Message Passings (MPs) is challenging due to the scaling complexity w.r.t. the number of nodes and over-smoothing. There has been growing interest in the community to introduce textitmulti-scale structures to GNNs for physical simulation. However, current state-of-the-art methods are limited by their reliance on the labor-intensive drawing of coarser meshes or building coarser levels based on spatial proximity, which can introduce wrong edges across geometry boundaries. Inspired by the bipartite graph determination, we propose a novel pooling strategy, textitbi-stride to tackle the aforementioned limitations. Bi-stride pools nodes on every other frontier of the breadth-first search (BFS), without the need for the manual drawing of coarser meshes and avoiding the wrong edges by spatial proximity. Additionally, it enables a one-MP scheme per level and non-parametrized pooling and unpooling by interpolations, resembling U-Nets, which significantly reduces computational costs. Experiments show that the proposed framework, textitBSMS-GNN, significantly outperforms existing methods in terms of both accuracy and computational efficiency in representative physical simulations.