mlsanityData to 7 Oct 2026

Simulating Continuum Mechanics with Multi-Scale Graph Neural Networks

arXiv 2106.04900 ↗First posted 9 Jun 2021cs.LG physics.flu-dyncited by 39

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Abstract (arXiv)

Continuum mechanics simulators, numerically solving one or more partial differential equations, are essential tools in many areas of science and engineering, but their performance often limits application in practice. Recent modern machine learning approaches have demonstrated their ability to accelerate spatio-temporal predictions, although, with only moderate accuracy in comparison. Here we introduce MultiScaleGNN, a novel multi-scale graph neural network model for learning to infer unsteady continuum mechanics. MultiScaleGNN represents the physical domain as an unstructured set of nodes, and it constructs one or more graphs, each of them encoding different scales of spatial resolution. Successive learnt message passing between these graphs improves the ability of GNNs to capture and forecast the system state in problems encompassing a range of length scales. Using graph representations, MultiScaleGNN can impose periodic boundary conditions as an inductive bias on the edges in the graphs, and achieve independence to the nodes' positions. We demonstrate this method on advection problems and incompressible fluid dynamics. Our results show that the proposed model can generalise from uniform advection fields to high-gradient fields on complex domains at test time and infer long-term Navier-Stokes solutions within a range of Reynolds numbers. Simulations obtained with MultiScaleGNN are between two and four orders of magnitude faster than the ones on which it was trained.

Key facts. Cited by 39 papers in the map (data through 7 Oct 2026). Builds on 25 of its 39 references that are in the map. First posted on arXiv 9 Jun 2021.

Builds on

25 of 39 references.

References · 25

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13 more
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 6; 2023: 10; 2024: 9; 2025: 8; 2026: 4.

39 papers.

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  1. 2022MultiScale MeshGraphNetscited by 94
  2. 2022HOOD: Hierarchical Graphs for Generalized Modelling of Clothing Dynamicscited by 77
  3. 2022Efficient Learning of Mesh-Based Physical Simulation with BSMS-GNNcited by 43
  4. 2022Learning to Solve PDE-constrained Inverse Problems with Graph Networkscited by 37
  5. 2024Probabilistic Weather Forecasting with Hierarchical Graph Neural Networkscited by 29
  6. 2023Graph-based Neural Weather Prediction for Limited Area Modelingcited by 22
  7. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  8. 2023MMGP: a Mesh Morphing Gaussian Process-based machine learning method for regression of physical problems under non-parameterized geometrical variabilitycited by 15
  9. 2021Differentiable Physics: A Position Piececited by 12
  10. 2023Multi-Scale Message Passing Neural PDE Solverscited by 12
  11. 2024Mesh-based Super-Resolution of Fluid Flows with Multiscale Graph Neural Networkscited by 10
  12. 2024Multi-Grid Graph Neural Networks with Self-Attention for Computational Mechanicscited by 9
27 more
  1. 2025MeshMask: Physics-Based Simulations with Masked Graph Neural Networkscited by 8
  2. 2024A Microstructure-based Graph Neural Network for Accelerating Multiscale Simulationscited by 7
  3. 2025AMR-Transformer: Enabling Efficient Long-range Interaction for Complex Neural Fluid Simulationcited by 7
  4. 2021Deep Surrogate for Direct Time Fluid Dynamicscited by 5
  5. 2023Multi-Scale Simulation of Complex Systems: A Perspective of Integrating Knowledge and Datacited by 5
  6. 2023Multi-GPU Approach for Training of Graph ML Models on large CFD Meshescited by 4
  7. 2025A graph neural network surrogate model for mesh-based crashworthiness prediction of vehicle panel componentscited by 4
  8. 2025Training Transformers for Mesh-Based Simulationscited by 4
  9. 2022REMuS-GNN: A Rotation-Equivariant Model for Simulating Continuum Dynamicscited by 3
  10. 2023An Implicit GNN Solver for Poisson-like problemscited by 3
  11. 2023Learning CO2 plume migration in faulted reservoirs with Graph Neural Networkscited by 3
  12. 2024Up-sampling-only and Adaptive Mesh-based GNN for Simulating Physical Systemscited by 3
  13. 2022DS-GPS : A Deep Statistical Graph Poisson Solver (for faster CFD simulations)cited by 2
  14. 2023On the Relationships between Graph Neural Networks for the Simulation of Physical Systems and Classical Numerical Methodscited by 2
  15. 2024LatticeGraphNet: A two-scale graph neural operator for simulating lattice structurescited by 2
  16. 2024Learning Physical Simulation with Message Passing Transformercited by 2
  17. 2024Scalable and Consistent Graph Neural Networks for Distributed Mesh-based Data-driven Modelingcited by 2
  18. 2023Proposing an intelligent mesh smoothing method with graph neural networkscited by 1
  19. 2025Curriculum Learning for Mesh-based simulationscited by 1
  20. 2025Graph Deep Learning for Intracranial Aneurysm Blood Flow Simulation and Risk Assessmentcited by 1
  21. 2024Statistical Edge Detection And UDF Learning For Shape Representationcited by 0
  22. 2025PLAID: A Unified Data Model for Machine Learning on Heterogeneous Physics Simulationscited by 0
  23. 2025Learning Deformable Body Interactions With Adaptive Spatial Tokenizationcited by 0
  24. 2026Deterministic and probabilistic neural surrogates of global hybrid-Vlasov simulationscited by 0
  25. 2026A Dual-Graph Spatiotemporal GNN Surrogate for Nonlinear Response Prediction of Reinforced Concrete Beams under Four-Point Bendingcited by 0
  26. 2026Physics-Informed Coarsening for Multigrid Graph Neural Surrogatescited by 0
  27. 2026Physics-Guided Dual-Stream Heterogeneous Graph Neural Network for Predicting Full-Field Structural Response of Stiffened Panelscited by 0