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Physics-informed neural networks (PINNs) for fluid mechanics: A review

arXiv 2105.09506 ↗First posted 20 May 2021physics.flu-dyn cs.LGcited by 470

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Abstract (arXiv)

Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier-Stokes equations (NSE), we still cannot incorporate seamlessly noisy data into existing algorithms, mesh-generation is complex, and we cannot tackle high-dimensional problems governed by parametrized NSE. Moreover, solving inverse flow problems is often prohibitively expensive and requires complex and expensive formulations and new computer codes. Here, we review flow physics-informed learning, integrating seamlessly data and mathematical models, and implementing them using physics-informed neural networks (PINNs). We demonstrate the effectiveness of PINNs for inverse problems related to three-dimensional wake flows, supersonic flows, and biomedical flows.

Key facts. Cited by 470 papers in the map (data through 7 Oct 2026). Builds on 27 of its 61 references that are in the map. First posted on arXiv 20 May 2021.

Builds on

27 of 61 references.

References · 27

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  3. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  4. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  5. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  6. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  7. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  8. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  9. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 203
  10. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  11. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  12. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
15 more
  1. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 139
  2. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 134
  3. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  4. 2019Transfer learning enhanced physics informed neural network for phase-field modeling of fracturecited by 116
  5. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  6. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  7. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  8. 2019Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networkscited by 76
  9. 2018Deep Learning of Vortex Induced Vibrationscited by 74
  10. 2017Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equationscited by 72
  11. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 63
  12. 2020NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation frameworkcited by 58
  13. 2020Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imagingcited by 40
  14. 2020Deep learning of free boundary and Stefan problemscited by 33
  15. 2021Physics-aware deep neural networks for surrogate modeling of turbulent natural convectioncited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 10; 2022: 56; 2023: 87; 2024: 125; 2025: 122; 2026: 70.

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  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  2. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  3. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstructioncited by 150
  4. 2023Scalable Transformer for PDE Surrogate Modelingcited by 123
  5. 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 103
  6. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 100
  7. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  8. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  9. 2023Operator Learning with Neural Fields: Tackling PDEs on General Geometriescited by 80
  10. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 78
  11. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  12. 2023Residual-based attention in physics-informed neural networkscited by 72
38 more
  1. 2022Human Trajectory Prediction via Neural Social Physicscited by 66
  2. 2022Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modelingcited by 65
  3. 2022Blending Neural Operators and Relaxation Methods in PDE Numerical Solverscited by 53
  4. 2022Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problemscited by 47
  5. 2021Physics informed neural networks for continuum micromechanicscited by 46
  6. 2023Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestrationcited by 46
  7. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  8. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 43
  9. 2022Machine Learning in Aerodynamic Shape Optimizationcited by 39
  10. 2023Learning Neural PDE Solvers with Parameter-Guided Channel Attentioncited by 37
  11. 2023Domain Agnostic Fourier Neural Operatorscited by 36
  12. 2021Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural networkcited by 35
  13. 2024Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problemscited by 35
  14. 2025T2VPhysBench: A First-Principles Benchmark for Physical Consistency in Text-to-Video Generationcited by 35
  15. 2022A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element methodcited by 33
  16. 2022NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operatorscited by 33
  17. 2023Correcting model misspecification in physics-informed neural networks (PINNs)cited by 33
  18. 2024Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?cited by 31
  19. 2023Super-Resolution Analysis via Machine Learning: A Survey for Fluid Flowscited by 30
  20. 2023L-HYDRA: Multi-Head Physics-Informed Neural Networkscited by 29
  21. 2022INO: Invariant Neural Operators for Learning Complex Physical Systems with Momentum Conservationcited by 28
  22. 2023Physics-Informed Computer Vision: A Review and Perspectivescited by 28
  23. 2022Bi-fidelity Modeling of Uncertain and Partially Unknown Systems using DeepONetscited by 27
  24. 2023Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorialcited by 27
  25. 2023Physics informed WNOcited by 26
  26. 2023Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regressioncited by 25
  27. 2022Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computingcited by 24
  28. 2023Deep Operator Network Approximation Rates for Lipschitz Operatorscited by 24
  29. 2023Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networkscited by 23
  30. 2024Generalizing Weather Forecast to Fine-grained Temporal Scales via Physics-AI Hybrid Modelingcited by 23
  31. 2023Harnessing the Power of Neural Operators with Automatically Encoded Conservation Lawscited by 22
  32. 2022Physics-informed deep-learning applications to experimental fluid mechanicscited by 21
  33. 2022Experience report of physics-informed neural networks in fluid simulations: pitfalls and frustrationcited by 21
  34. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  35. 2023Stacked networks improve physics-informed training: applications to neural networks and deep operator networkscited by 21
  36. 2022Fully probabilistic deep models for forward and inverse problems in parametric PDEscited by 19
  37. 2024DiffuSolve: Diffusion-based Solver for Non-convex Trajectory Optimizationcited by 19
  38. 2023Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapescited by 18

50 most cited of 470 All 470 (JSON)