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

arXiv: https://arxiv.org/abs/2107.10711 · HTML: https://mlsanity.com/paper/2107.10711/

## 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.

## Builds on (most cited first, 3 of 3)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Convolutional-network models to predict wall-bounded turbulence from wall quantities](https://mlsanity.com/paper/2006.12483/) (2020)

## Built on by (most cited first, 30 of 68)

- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective](https://mlsanity.com/paper/2502.00604/) (2025)
- [HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks](https://mlsanity.com/paper/2111.01008/) (2021)
- [DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectories](https://mlsanity.com/paper/2202.07176/) (2022)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (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)
- [Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs)](https://mlsanity.com/paper/2209.09988/) (2022)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systems](https://mlsanity.com/paper/2301.10391/) (2023)
- [Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraints](https://mlsanity.com/paper/2507.08124/) (2025)
- [Simulating Three-dimensional Turbulence with Physics-informed Neural Networks](https://mlsanity.com/paper/2507.08972/) (2025)
- [Predicting the temporal dynamics of turbulent channels through deep learning](https://mlsanity.com/paper/2203.00974/) (2022)
- [Thermodynamics-informed neural networks for physically realistic mixed reality](https://mlsanity.com/paper/2210.13414/) (2022)
- [Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networks](https://mlsanity.com/paper/2403.03223/) (2024)
- [Physics-informed neural network solution of thermo-hydro-mechanical (THM) processes in porous media](https://mlsanity.com/paper/2203.01514/) (2022)
- [VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Models](https://mlsanity.com/paper/2506.13754/) (2025)
- [Physics-informed PointNet: On how many irregular geometries can it solve an inverse problem simultaneously? Application to linear elasticity](https://mlsanity.com/paper/2303.13634/) (2023)
- [Learning Hamiltonian neural Koopman operator and simultaneously sustaining and discovering conservation law](https://mlsanity.com/paper/2406.02154/) (2024)
- [NeuroSEM: A hybrid framework for simulating multiphysics problems by coupling PINNs and spectral elements](https://mlsanity.com/paper/2407.21217/) (2024)
- [Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective](https://mlsanity.com/paper/2506.19805/) (2025)
- [RANS-PINN based Simulation Surrogates for Predicting Turbulent Flows](https://mlsanity.com/paper/2306.06034/) (2023)
- [Approximating Families of Sharp Solutions to Fisher's Equation with Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.08313/) (2024)
- [FastVPINNs: Tensor-Driven Acceleration of VPINNs for Complex Geometries](https://mlsanity.com/paper/2404.12063/) (2024)
- [Enhancing Graph U-Nets for Mesh-Agnostic Spatio-Temporal Flow Prediction](https://mlsanity.com/paper/2406.03789/) (2024)
- [Machine Learning Applications to Computational Plasma Physics and Reduced-Order Plasma Modeling: A Perspective](https://mlsanity.com/paper/2409.02349/) (2024)
- [EPINN-NSE: Enhanced Physics-Informed Neural Networks for Solving Navier-Stokes Equations](https://mlsanity.com/paper/2304.03689/) (2023)
- [Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology](https://mlsanity.com/paper/2408.07201/) (2024)

## 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.