mlsanityData to 7 Oct 2026

Machine learning accelerated computational fluid dynamics

arXiv 2102.01010 ↗First posted 28 Jan 2021physics.flu-dyn cs.LGcited by 461

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Cited by 27 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

Numerical simulation of fluids plays an essential role in modeling many physical phenomena, such as weather, climate, aerodynamics and plasma physics. Fluids are well described by the Navier-Stokes equations, but solving these equations at scale remains daunting, limited by the computational cost of resolving the smallest spatiotemporal features. This leads to unfavorable trade-offs between accuracy and tractability. Here we use end-to-end deep learning to improve approximations inside computational fluid dynamics for modeling two-dimensional turbulent flows. For both direct numerical simulation of turbulence and large eddy simulation, our results are as accurate as baseline solvers with 8-10x finer resolution in each spatial dimension, resulting in 40-80x fold computational speedups. Our method remains stable during long simulations, and generalizes to forcing functions and Reynolds numbers outside of the flows where it is trained, in contrast to black box machine learning approaches. Our approach exemplifies how scientific computing can leverage machine learning and hardware accelerators to improve simulations without sacrificing accuracy or generalization.

Key facts. Cited by 461 papers in the map (data through 7 Oct 2026). Builds on 22 of its 62 references that are in the map. First posted on arXiv 28 Jan 2021.

Builds on

22 of 62 references.

References · 22

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10 more
  1. 2019DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)cited by 40
  2. 2020CFDNet: a deep learning-based accelerator for fluid simulationscited by 40
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 41; 2022: 76; 2023: 94; 2024: 73; 2025: 84; 2026: 93.

461 papers.

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  1. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  2. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  3. 2023ClimaX: A foundation model for weather and climatecited by 325
  4. 2021Modern Koopman Theory for Dynamical Systemscited by 269
  5. 2021Factorized Fourier Neural Operatorscited by 237
  6. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  7. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  8. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 205
  9. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
  10. 2022Forecasting Global Weather with Graph Neural Networkscited by 165
  11. 2022Respecting causality is all you need for training physics-informed neural networkscited by 154
  12. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstructioncited by 150
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  1. 2021Enhancing Computational Fluid Dynamics with Machine Learningcited by 137
  2. 2023Scalable Transformer for PDE Surrogate Modelingcited by 123
  3. 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 119
  4. 2023DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecastingcited by 105
  5. 2021Deep Generative Models in Engineering Design: A Reviewcited by 103
  6. 2024Rolling Diffusion Modelscited by 102
  7. 2022Monarch: Expressive Structured Matrices for Efficient and Accurate Trainingcited by 100
  8. 2023In-Context Operator Learning with Data Prompts for Differential Equation Problemscited by 98
  9. 2022MultiScale MeshGraphNetscited by 94
  10. 2022Clifford Neural Layers for PDE Modelingcited by 91
  11. 2023Score-based Data Assimilationcited by 91
  12. 2021Learned Coarse Models for Efficient Turbulence Simulationcited by 90
  13. 2023Multiple Physics Pretraining for Physical Surrogate Modelscited by 90
  14. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 87
  15. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  16. 2021Gradients are Not All You Needcited by 82
  17. 2022Predicting Physics in Mesh-reduced Space with Temporal Attentioncited by 82
  18. 2024ClimODE: Climate and Weather Forecasting with Physics-informed Neural ODEscited by 78
  19. 2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 76
  20. 2022Continuous PDE Dynamics Forecasting with Implicit Neural Representationscited by 71
  21. 2024PirateNets: Physics-informed Deep Learning with Residual Adaptive Networkscited by 70
  22. 2022SocialVAE: Human Trajectory Prediction using Timewise Latentscited by 65
  23. 2023Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulationcited by 65
  24. 2023Separable Physics-Informed Neural Networkscited by 64
  25. 2021Encoding physics to learn reaction-diffusion processescited by 61
  26. 2024On conditional diffusion models for PDE simulationscited by 56
  27. 2022Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equationscited by 55
  28. 2023Towards Stability of Autoregressive Neural Operatorscited by 54
  29. 2022Blending Neural Operators and Relaxation Methods in PDE Numerical Solverscited by 53
  30. 2024ACE2: Accurately learning subseasonal to decadal atmospheric variability and forced responsescited by 53
  31. 2022Generalizing to New Physical Systems via Context-Informed Dynamics Modelcited by 49
  32. 2022Learning to correct spectral methods for simulating turbulent flowscited by 49
  33. 2023Multi-stage Neural Networks: Function Approximator of Machine Precisioncited by 48
  34. 2023From Zero to Turbulence: Generative Modeling for 3D Flow Simulationcited by 46
  35. 2022Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamicscited by 44
  36. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 43
  37. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 42
  38. 2023CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamicscited by 40

50 most cited of 461 All 461 (JSON)