# Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations

arXiv: https://arxiv.org/abs/2206.09571 · HTML: https://mlsanity.com/paper/2206.09571/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 57 references that are in the map.
- First posted on arXiv 20 Jun 2022.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Exploring Simple Siamese Representation Learning](https://mlsanity.com/paper/2011.10566/) (2020)
- [Deep Reinforcement Learning with Double Q-learning](https://mlsanity.com/paper/1509.06461/) (2015)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains](https://mlsanity.com/paper/2007.11207/) (2020)
- [Meta-learning PINN loss functions](https://mlsanity.com/paper/2107.05544/) (2021)
- [DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation](https://mlsanity.com/paper/2012.02681/) (2020)
- [Physics guided machine learning using simplified theories](https://mlsanity.com/paper/2012.13343/) (2020)
- [Linearly Constrained Neural Networks](https://mlsanity.com/paper/2002.01600/) (2020)

## Built on by (most cited first, 5 of 5)

- [Deciphering and integrating invariants for neural operator learning with various physical mechanisms](https://mlsanity.com/paper/2311.14361/) (2023)
- [Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation](https://mlsanity.com/paper/2302.05104/) (2023)
- [Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs](https://mlsanity.com/paper/2303.11205/) (2023)
- [Neural Networks-based Random Vortex Methods for Modelling Incompressible Flows](https://mlsanity.com/paper/2405.13691/) (2024)
- [OmniFluids: Physics Pre-trained Modeling of Fluid Dynamics](https://mlsanity.com/paper/2506.10862/) (2025)

## Abstract (arXiv)

Navier-Stokes equations are significant partial differential equations that describe the motion of fluids such as liquids and air. Due to the importance of Navier-Stokes equations, the development on efficient numerical schemes is important for both science and engineer. Recently, with the development of AI techniques, several approaches have been designed to integrate deep neural networks in simulating and inferring the fluid dynamics governed by incompressible Navier-Stokes equations, which can accelerate the simulation or inferring process in a mesh-free and differentiable way. In this paper, we point out that the capability of existing deep Navier-Stokes informed methods is limited to handle non-smooth or fractional equations, which are two critical situations in reality. To this end, we propose the Deep Random Vortex Method (DRVM), which combines the neural network with a random vortex dynamics system equivalent to the Navier-Stokes equation. Specifically, the random vortex dynamics motivates a Monte Carlo based loss function for training the neural network, which avoids the calculation of derivatives through auto-differentiation. Therefore, DRVM not only can efficiently solve Navier-Stokes equations involving rough path, non-differentiable initial conditions and fractional operators, but also inherits the mesh-free and differentiable benefits of the deep-learning-based solver. We conduct experiments on the Cauchy problem, parametric solver learning, and the inverse problem of both 2-d and 3-d incompressible Navier-Stokes equations. The proposed method achieves accurate results for simulation and inference of Navier-Stokes equations. Especially for the cases that include singular initial conditions, DRVM significantly outperforms existing PINN method.