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Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equations

arXiv 2107.10711 ↗First posted 22 Jul 2021physics.flu-dyn cs.LG physics.comp-phcited by 68

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Abstract (arXiv)

Physics-informed neural networks (PINNs) are successful machine-learning methods for the solution and identification of partial differential equations (PDEs). We employ PINNs for solving the Reynolds-averaged Navierunicodex2013Stokes (RANS) equations for incompressible turbulent flows without any specific model or assumption for turbulence, and by taking only the data on the domain boundaries. We first show the applicability of PINNs for solving the Navierunicodex2013Stokes equations for laminar flows by solving the Falknerunicodex2013Skan boundary layer. We then apply PINNs for the simulation of four turbulent-flow cases, i.e., zero-pressure-gradient boundary layer, adverse-pressure-gradient boundary layer, and turbulent flows over a NACA4412 airfoil and the periodic hill. Our results show the excellent applicability of PINNs for laminar flows with strong pressure gradients, where predictions with less than 1% error can be obtained. For turbulent flows, we also obtain very good accuracy on simulation results even for the Reynolds-stress components.

Key facts. Cited by 68 papers in the map (data through 14 Oct 2026). Builds on 3 of its 27 references that are in the map. First posted on arXiv 22 Jul 2021.

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3 of 27 references.

References · 3

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2015Automatic differentiation in machine learning: a surveycited by 855
  3. 2020Convolutional-network models to predict wall-bounded turbulence from wall quantitiescited by 47

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 2; 2022: 11; 2023: 8; 2024: 21; 2025: 18; 2026: 8.

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  1. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 413
  2. 2021Enhancing Computational Fluid Dynamics with Machine Learningcited by 138
  3. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 108
  4. 2025Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspectivecited by 56
  5. 2021HyperPINN: Learning parameterized differential equations with physics-informed hypernetworkscited by 39
  6. 2022DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectoriescited by 30
  7. 2022Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computingcited by 24
  8. 2023Machine Learning for Partial Differential Equationscited by 21
  9. 2022Improving aircraft performance using machine learning: a reviewcited by 19
  10. 2022Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs)cited by 17
  11. 2024A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domainscited by 15
  12. 2023Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systemscited by 14
38 more
  1. 2025Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraintscited by 14
  2. 2025Simulating Three-dimensional Turbulence with Physics-informed Neural Networkscited by 14
  3. 2022Predicting the temporal dynamics of turbulent channels through deep learningcited by 10
  4. 2022Thermodynamics-informed neural networks for physically realistic mixed realitycited by 9
  5. 2024Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networkscited by 9
  6. 2022Physics-informed neural network solution of thermo-hydro-mechanical (THM) processes in porous mediacited by 8
  7. 2025VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Modelscited by 8
  8. 2023Physics-informed PointNet: On how many irregular geometries can it solve an inverse problem simultaneously? Application to linear elasticitycited by 7
  9. 2024Learning Hamiltonian neural Koopman operator and simultaneously sustaining and discovering conservation lawcited by 7
  10. 2024NeuroSEM: A hybrid framework for simulating multiphysics problems by coupling PINNs and spectral elementscited by 6
  11. 2025Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspectivecited by 6
  12. 2023RANS-PINN based Simulation Surrogates for Predicting Turbulent Flowscited by 5
  13. 2024Approximating Families of Sharp Solutions to Fisher's Equation with Physics-Informed Neural Networkscited by 5
  14. 2024FastVPINNs: Tensor-Driven Acceleration of VPINNs for Complex Geometriescited by 5
  15. 2024Enhancing Graph U-Nets for Mesh-Agnostic Spatio-Temporal Flow Predictioncited by 4
  16. 2024Machine Learning Applications to Computational Plasma Physics and Reduced-Order Plasma Modeling: A Perspectivecited by 4
  17. 2023EPINN-NSE: Enhanced Physics-Informed Neural Networks for Solving Navier-Stokes Equationscited by 3
  18. 2024Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiologycited by 3
  19. 2024Mean flow data assimilation using physics-constrained Graph Neural Networkscited by 3
  20. 2025Comparison of CNN-based deep learning architectures for unsteady CFD acceleration on small datasetscited by 3
  21. 2025Physics-informed KAN PointNet: Deep learning for simultaneous solutions to inverse problems in incompressible flow on numerous irregular geometriescited by 3
  22. 2025EquiNO: A Physics-Informed Neural Operator for Multiscale Simulationscited by 3
  23. 2025Enforcing hidden physics in physics-informed neural networkscited by 3
  24. 2023Reconstructing Rayleigh-Benard flows out of temperature-only measurements using Physics-Informed Neural Networkscited by 2
  25. 2023Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUpcited by 2
  26. 2024EngineBench: Flow Reconstruction in the Transparent Combustion Chamber III Optical Enginecited by 2
  27. 2025A discrete physics-informed training for projection-based reduced order models with neural networkscited by 2
  28. 2025Hierarchical-embedding autoencoder with a predictor (HEAP) as efficient architecture for learning long-term evolution of complex multi-scale physical systemscited by 2
  29. 2025Noisy PDE Training Requires Bigger PINNscited by 2
  30. 2025Convergence of Stochastic Gradient Methods for Wide Two-Layer Physics-Informed Neural Networks for the Poisson Equationcited by 2
  31. 2026ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidancecited by 2
  32. 2026Ambient Physics: Training Neural PDE Solvers with Partial Observationscited by 2
  33. 2022Physics-informed Neural Networks approach to solve the Blasius functioncited by 1
  34. 2024FlamePINN-1D: Physics-informed neural networks to solve forward and inverse problems of 1D laminar flamescited by 1
  35. 2024Using Parametric PINNs for Predicting Internal and External Turbulent Flowscited by 1
  36. 2025Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networkscited by 1
  37. 2025RamPINN: Recovering Raman Spectra From Coherent Anti-Stokes Spectra Using Embedded Physicscited by 1
  38. 2026On the Role of Consistency Between Physics and Data in Physics-Informed Neural Networkscited by 1

50 most cited of 68 All 68 (JSON)