# Enhancing Computational Fluid Dynamics with Machine Learning

arXiv: https://arxiv.org/abs/2110.02085 · HTML: https://mlsanity.com/paper/2110.02085/

## Key facts

- Cited by 138 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 156 references that are in the map.
- First posted on arXiv 5 Oct 2021.

## Builds on (most cited first, 30 of 35)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [A Tour of Reinforcement Learning: The View from Continuous Control](https://mlsanity.com/paper/1806.09460/) (2018)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Machine learning for molecular simulation](https://mlsanity.com/paper/1911.02792/) (2019)
- [The role of artificial intelligence in achieving the Sustainable Development Goals](https://mlsanity.com/paper/1905.00501/) (2019)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (2020)
- [Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equations](https://mlsanity.com/paper/2107.10711/) (2021)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?](https://mlsanity.com/paper/2103.09655/) (2021)
- [Scientific multi-agent reinforcement learning for wall-models of turbulent flows](https://mlsanity.com/paper/2106.11144/) (2021)
- [Convolutional-network models to predict wall-bounded turbulence from wall quantities](https://mlsanity.com/paper/2006.12483/) (2020)
- [Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows](https://mlsanity.com/paper/2109.01514/) (2021)
- [CFDNet: a deep learning-based accelerator for fluid simulations](https://mlsanity.com/paper/2005.04485/) (2020)
- [Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms](https://mlsanity.com/paper/2002.09726/) (2020)
- [Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh](https://mlsanity.com/paper/1910.08613/) (2019)
- [Interpretable deep-learning models to help achieve the Sustainable Development Goals](https://mlsanity.com/paper/2108.10744/) (2021)
- [Enhancement of shock-capturing methods via machine learning](https://mlsanity.com/paper/2002.02521/) (2020)

## Built on by (most cited first, 30 of 138)

- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [β-Variational autoencoders and transformers for reduced-order modelling of fluid flows](https://mlsanity.com/paper/2304.03571/) (2023)
- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fields](https://mlsanity.com/paper/2406.02176/) (2024)
- [Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning](https://mlsanity.com/paper/2402.15734/) (2024)
- [Self-Supervised Learning with Lie Symmetries for Partial Differential Equations](https://mlsanity.com/paper/2307.05432/) (2023)
- [Bayesian Conditional Diffusion Models for Versatile Spatiotemporal Turbulence Generation](https://mlsanity.com/paper/2311.07896/) (2023)
- [The transformative potential of machine learning for experiments in fluid mechanics](https://mlsanity.com/paper/2303.15832/) (2023)
- [Universal Physics Transformers: A Framework For Efficiently Scaling Neural Operators](https://mlsanity.com/paper/2402.12365/) (2024)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2312.05320/) (2023)
- [Physics-informed deep-learning applications to experimental fluid mechanics](https://mlsanity.com/paper/2203.15402/) (2022)
- [Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2303.17078/) (2023)
- [Improving aircraft performance using machine learning: a review](https://mlsanity.com/paper/2210.11481/) (2022)
- [SuperBench: A Super-Resolution Benchmark Dataset for Scientific Machine Learning](https://mlsanity.com/paper/2306.14070/) (2023)
- [Earthfarseer: Versatile Spatio-Temporal Dynamical Systems Modeling in One Model](https://mlsanity.com/paper/2312.08403/) (2023)
- [AhmedML: High-Fidelity Computational Fluid Dynamics Dataset for Incompressible, Low-Speed Bluff Body Aerodynamics](https://mlsanity.com/paper/2407.20801/) (2024)
- [Residual-based physics-informed transfer learning: A hybrid method for accelerating long-term CFD simulations via deep learning](https://mlsanity.com/paper/2206.06817/) (2022)
- [Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator](https://mlsanity.com/paper/2408.02965/) (2024)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [XLB: A differentiable massively parallel lattice Boltzmann library in Python](https://mlsanity.com/paper/2311.16080/) (2023)
- [The impact of internal variability on benchmarking deep learning climate emulators](https://mlsanity.com/paper/2408.05288/) (2024)
- [NeuralDEM -- Real-time Simulation of Industrial Particulate Flows](https://mlsanity.com/paper/2411.09678/) (2024)
- [Local neural operator for solving transient partial differential equations on varied domains](https://mlsanity.com/paper/2203.08145/) (2022)
- [Inverse Problems with Diffusion Models: A MAP Estimation Perspective](https://mlsanity.com/paper/2407.20784/) (2024)
- [Emerging trends in machine learning for computational fluid dynamics](https://mlsanity.com/paper/2211.15145/) (2022)
- [Stochastic Latent Transformer: Efficient Modelling of Stochastically Forced Zonal Jets](https://mlsanity.com/paper/2310.16741/) (2023)
- [A conditional latent autoregressive recurrent model for generation and forecasting of beam dynamics in particle accelerators](https://mlsanity.com/paper/2403.13858/) (2024)

## Abstract (arXiv)

Machine learning is rapidly becoming a core technology for scientific computing, with numerous opportunities to advance the field of computational fluid dynamics. In this Perspective, we highlight some of the areas of highest potential impact, including to accelerate direct numerical simulations, to improve turbulence closure modeling, and to develop enhanced reduced-order models. We also discuss emerging areas of machine learning that are promising for computational fluid dynamics, as well as some potential limitations that should be taken into account.