Deep Random Vortex Method for Simulation and Inference of Navier-Stokes
Equations
- URL: http://arxiv.org/abs/2206.09571v1
- Date: Mon, 20 Jun 2022 04:58:09 GMT
- Title: Deep Random Vortex Method for Simulation and Inference of Navier-Stokes
Equations
- Authors: Rui Zhang, Peiyan Hu, Qi Meng, Yue Wang, Rongchan Zhu, Bingguang Chen,
Zhi-Ming Ma, Tie-Yan Liu
- Abstract summary: Navier-Stokes equations are significant partial differential equations that describe the motion of fluids such as liquids and air.
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.
We propose the emphDeep Random Vortex Method (DRVM), which combines the neural network with a random vortex dynamics system equivalent to the Navier-Stokes equation.
- Score: 69.5454078868963
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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