mlsanityData to 14 Oct 2026

Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction

arXiv 2007.04439 ↗First posted 8 Jul 2020cs.LG physics.comp-ph stat.MLcited by 138

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Abstract (arXiv)

Solving large complex partial differential equations (PDEs), such as those that arise in computational fluid dynamics (CFD), is a computationally expensive process. This has motivated the use of deep learning approaches to approximate the PDE solutions, yet the simulation results predicted from these approaches typically do not generalize well to truly novel scenarios. In this work, we develop a hybrid (graph) neural network that combines a traditional graph convolutional network with an embedded differentiable fluid dynamics simulator inside the network itself. By combining an actual CFD simulator (run on a much coarser resolution representation of the problem) with the graph network, we show that we can both generalize well to new situations and benefit from the substantial speedup of neural network CFD predictions, while also substantially outperforming the coarse CFD simulation alone.

Key facts. Cited by 138 papers in the map (data through 14 Oct 2026). First posted on arXiv 8 Jul 2020.

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Citing papers per year: 2021: 27; 2022: 31; 2023: 29; 2024: 24; 2025: 13; 2026: 14.

138 papers.

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  1. 2022Message Passing Neural PDE Solverscited by 386
  2. 2022Transformer for Partial Differential Equations' Operator Learningcited by 283
  3. 2021GRAND: Graph Neural Diffusioncited by 258
  4. 2021PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equationscited by 106
  5. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 98
  6. 2022MultiScale MeshGraphNetscited by 98
  7. 2021Learned Coarse Models for Efficient Turbulence Simulationcited by 91
  8. 2021Beltrami Flow and Neural Diffusion on Graphscited by 84
  9. 2022Continuous PDE Dynamics Forecasting with Implicit Neural Representationscited by 73
  10. 2023Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulationcited by 71
  11. 2022When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learningcited by 65
  12. 2021Encoding physics to learn reaction-diffusion processescited by 61
38 more
  1. 2021Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modelingcited by 59
  2. 2022Efficient Learning of Mesh-Based Physical Simulation with BSMS-GNNcited by 45
  3. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 44
  4. 2021Simulating Continuum Mechanics with Multi-Scale Graph Neural Networkscited by 40
  5. 2021Reinforcement Learning for Adaptive Mesh Refinementcited by 39
  6. 2021Efficient Differentiable Simulation of Articulated Bodiescited by 39
  7. 2022Differentiable Simulation of Soft Multi-body Systemscited by 39
  8. 2023Learning Neural PDE Solvers with Parameter-Guided Channel Attentioncited by 38
  9. 2022Physical Design using Differentiable Learned Simulatorscited by 37
  10. 2022Learning to Solve PDE-constrained Inverse Problems with Graph Networkscited by 37
  11. 2022Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networkscited by 34
  12. 2021Graph Neural Networks Accelerated Molecular Dynamicscited by 33
  13. 2021Constraint-based graph network simulatorcited by 27
  14. 2021Noether Networks: Meta-Learning Useful Conserved Quantitiescited by 26
  15. 2025Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networkscited by 25
  16. 2022Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learningcited by 24
  17. 2024BENO: Boundary-embedded Neural Operators for Elliptic PDEscited by 24
  18. 2021PhysGNN: A Physics-Driven Graph Neural Network Based Model for Predicting Soft Tissue Deformation in Image-Guided Neurosurgerycited by 22
  19. 2022Improving Graph Neural Networks with Learnable Propagation Operatorscited by 19
  20. 2023Multiscale Graph Neural Network Autoencoders for Interpretable Scientific Machine Learningcited by 18
  21. 2023Learning Flexible Body Collision Dynamics with Hierarchical Contact Mesh Transformercited by 18
  22. 2021Conditionally Parameterized, Discretization-Aware Neural Networks for Mesh-Based Modeling of Physical Systemscited by 17
  23. 2022Learning the Solution Operator of Boundary Value Problems using Graph Neural Networkscited by 17
  24. 2023Swarm Reinforcement Learning For Adaptive Mesh Refinementcited by 17
  25. 2021A Deep Emulator for Secondary Motion of 3D Characterscited by 16
  26. 2022M2N: Mesh Movement Networks for PDE Solverscited by 16
  27. 2023Grounding Graph Network Simulators using Physical Sensor Observationscited by 15
  28. 2023Surrogate Modeling of Car Drag Coefficient with Depth and Normal Renderingscited by 15
  29. 2023Feature Transportation Improves Graph Neural Networkscited by 15
  30. 2022A Finite Element-Inspired Hypergraph Neural Network: Application to Fluid Dynamics Simulationscited by 14
  31. 2023Multi-scale Time-stepping of Partial Differential Equations with Transformerscited by 14
  32. 2024A review of graph neural network applications in mechanics-related domainscited by 14
  33. 2025Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networkscited by 14
  34. 2021Differentiable Physics: A Position Piececited by 12
  35. 20233D-IntPhys: Towards More Generalized 3D-grounded Visual Intuitive Physics under Challenging Scenescited by 12
  36. 2025Machine learning for modelling unstructured grid data in computational physics: a reviewcited by 12
  37. 2025UniPhy: Learning a Unified Constitutive Model for Inverse Physics Simulationcited by 12
  38. 2021Direct Prediction of Steady-State Flow Fields in Meshed Domain with Graph Networkscited by 11

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