# Learning solutions of parametric Navier-Stokes with physics-informed neural networks

arXiv: https://arxiv.org/abs/2402.03153 · HTML: https://mlsanity.com/paper/2402.03153/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 23 references that are in the map.
- First posted on arXiv 5 Feb 2024.

## Builds on (most cited first, 15 of 15)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data](https://mlsanity.com/paper/1808.04327/) (2018)
- [A novel meta-learning initialization method for physics-informed neural networks](https://mlsanity.com/paper/2107.10991/) (2021)
- [A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs](https://mlsanity.com/paper/2110.13361/) (2021)
- [An extended physics informed neural network for preliminary analysis of parametric optimal control problems](https://mlsanity.com/paper/2110.13530/) (2021)

## Built on by (most cited first, 4 of 4)

- [Physics-informed Discretization-independent Deep Compositional Operator Network](https://mlsanity.com/paper/2404.13646/) (2024)
- [Unraveling particle dark matter with Physics-Informed Neural Networks](https://mlsanity.com/paper/2502.17597/) (2025)
- [Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networks](https://mlsanity.com/paper/2602.21988/) (2026)
- [Hybrid Joint-Selective Optimization: Reduced-Space Levenberg-Marquardt Refinement of Low-Dimensional Parameters of Interest](https://mlsanity.com/paper/2609.37308/) (2026)

## Abstract (arXiv)

We leverage Physics-Informed Neural Networks (PINNs) to learn solution functions of parametric Navier-Stokes Equations (NSE). Our proposed approach results in a feasible optimization problem setup that bypasses PINNs' limitations in converging to solutions of highly nonlinear parametric-PDEs like NSE. We consider the parameter(s) of interest as inputs of PINNs along with spatio-temporal coordinates, and train PINNs on generated numerical solutions of parametric-PDES for instances of the parameters. We perform experiments on the classical 2D flow past cylinder problem aiming to learn velocities and pressure functions over a range of Reynolds numbers as parameter of interest. Provision of training data from generated numerical simulations allows for interpolation of the solution functions for a range of parameters. Therefore, we compare PINNs with unconstrained conventional Neural Networks (NN) on this problem setup to investigate the effectiveness of considering the PDEs regularization in the loss function. We show that our proposed approach results in optimizing PINN models that learn the solution functions while making sure that flow predictions are in line with conservational laws of mass and momentum. Our results show that PINN results in accurate prediction of gradients compared to NN model, this is clearly visible in predicted vorticity fields given that none of these models were trained on vorticity labels.