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

TL;DR

This paper introduces the Deep Random Vortex Method (DRVM), a novel neural network approach that efficiently simulates and infers Navier-Stokes equations, especially in challenging non-smooth and fractional cases, outperforming existing methods.

Contribution

The paper proposes DRVM, combining neural networks with vortex dynamics and Monte Carlo loss, enabling mesh-free, differentiable solutions for complex Navier-Stokes problems with rough and fractional operators.

Findings

DRVM accurately simulates 2D and 3D Navier-Stokes equations.

DRVM outperforms PINN in cases with singular initial conditions.

The method effectively handles non-differentiable and fractional operators.

Abstract

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 \emph{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.