# Applying Machine Learning to Study Fluid Mechanics

arXiv: https://arxiv.org/abs/2110.02083 · HTML: https://mlsanity.com/paper/2110.02083/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 113 references that are in the map.
- First posted on arXiv 5 Oct 2021.

## Builds on (most cited first, 30 of 32)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (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)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [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)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Techniques for Interpretable Machine Learning](https://mlsanity.com/paper/1808.00033/) (2018)
- [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)
- [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)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [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)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [Efficient collective swimming by harnessing vortices through deep reinforcement learning](https://mlsanity.com/paper/1802.02674/) (2018)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017)
- [Learning Symbolic Physics with Graph Networks](https://mlsanity.com/paper/1909.05862/) (2019)
- [Relevance of Rotationally Equivariant Convolutions for Predicting Molecular Properties](https://mlsanity.com/paper/2008.08461/) (2020)
- [Unsupervised Learning of Lagrangian Dynamics from Images for Prediction and Control](https://mlsanity.com/paper/2007.01926/) (2020)
- [Enhancement of shock-capturing methods via machine learning](https://mlsanity.com/paper/2002.02521/) (2020)

## Built on by (most cited first, 13 of 13)

- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [A multifidelity approach to continual learning for physical systems](https://mlsanity.com/paper/2304.03894/) (2023)
- [XLB: A differentiable massively parallel lattice Boltzmann library in Python](https://mlsanity.com/paper/2311.16080/) (2023)
- [Neural Autoencoder-Based Structure-Preserving Model Order Reduction and Control Design for High-Dimensional Physical Systems](https://mlsanity.com/paper/2312.06256/) (2023)
- [How to Control Hydrodynamic Force on Fluidic Pinball via Deep Reinforcement Learning](https://mlsanity.com/paper/2304.11526/) (2023)
- [Additive-feature-attribution methods: a review on explainable artificial intelligence for fluid dynamics and heat transfer](https://mlsanity.com/paper/2409.11992/) (2024)
- [Rotationally Equivariant Super-Resolution of Velocity Fields in Two-Dimensional Fluids Using Convolutional Neural Networks](https://mlsanity.com/paper/2202.11099/) (2022)
- [Physics-Informed Machine Learning of Argon Gas-Driven Melt Pool Dynamics](https://mlsanity.com/paper/2307.12304/) (2023)
- [Deep convolutional surrogates and degrees of freedom in thermal design](https://mlsanity.com/paper/2208.07482/) (2022)
- [A Kronecker product accelerated efficient sparse Gaussian Process (E-SGP) for flow emulation](https://mlsanity.com/paper/2312.10023/) (2023)
- [Towards a Real-Time Simulation of Elastoplastic Deformation Using Multi-Task Neural Networks](https://mlsanity.com/paper/2411.05575/) (2024)
- [Upstream flow geometries can be uniquely learnt from single-point turbulence signatures](https://mlsanity.com/paper/2412.10630/) (2024)
- [Physics-guided surrogate learning enables zero-shot control of turbulent wings](https://mlsanity.com/paper/2604.09434/) (2026)

## Abstract (arXiv)

This paper provides a short overview of how to use machine learning to build data-driven models in fluid mechanics. The process of machine learning is broken down into five stages: (1) formulating a problem to model, (2) collecting and curating training data to inform the model, (3) choosing an architecture with which to represent the model, (4) designing a loss function to assess the performance of the model, and (5) selecting and implementing an optimization algorithm to train the model. At each stage, we discuss how prior physical knowledge may be embedding into the process, with specific examples from the field of fluid mechanics.