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.
翻译:导航- Stokes 方程式是描述液体和空气等流体运动的重要部分差异方程式。 由于 Navier- Stokes 方程式的重要性, 高效数字方案的发展对科学和工程师都很重要。 最近, 随着AI 技术的发展, 设计了几种方法, 将深神经网络整合到模拟和推断由不压缩的 Navier- Stokes 方程式所支配的流体动态中, 这可以加速模拟或推导过程, 以无线和不同的方式。 在本文中, 我们指出, 现有的深层 Navier- Stokes 方程式对等方程式的重要性, 现有的深纳维- Stokes 方程式对于处理非moot- Stokes 方程式的重要性非常重要。 我们为此建议采用 \ emph{ epormation Vortexex 法} (DRVMM), 将神经网络与随机的离流电动力动态系统结合起来, 与 Navier- Stokes 平面方程式相当。 具体而言, 随机变式的解- solent- demoke- demoke- demotional- demotional- demodeal- dal- demodal- demodal- demotional- demotionalalal- demotional- demodalalal commodal commotional commodal comtional commodal- commotional commotional commotional commotional commotional commotional comtional commods commods commodal ets commods commodess commodess commodal commodal commodal commal ets ets ets ors ors ets etdaldaldaldalds etdaldaldalds comms ets et et, ets ets ets etdaldal etaldaldaldaldaldaldaldaldaldal etaldaldaldal