In this paper we introduce a novel Neural Networks-based approach for approximating solutions to the (2D) incompressible Navier--Stokes equations, which is an extension of so called Deep Random Vortex Methods (DRVM), that does not require the knowledge of the Biot--Savart kernel associated to the computational domain. Our algorithm uses a Neural Network (NN), that approximates the vorticity based on a loss function that uses a computationally efficient formulation of the Random Vortex Dynamics. The neural vorticity estimator is then combined with traditional numerical PDE-solvers, which can be considered as a final implicit linear layer of the network, for the Poisson equation to compute the velocity field. The main advantage of our method compared to the standard DRVM and other NN-based numerical algorithms is that it strictly enforces physical properties, such as incompressibility or (no slip) boundary conditions, which might be hard to guarantee otherwise. The approximation abilities of our algorithm, and its capability for incorporating measurement data, are validated by several numerical experiments.

翻译：本文提出一种基于神经网络的新方法，用于近似求解二维不可压缩Navier-Stokes方程。该方法是对所谓深度随机涡方法(DRVM)的扩展，无需已知计算域对应的Biot-Savart核函数。我们的算法采用神经网络，通过基于计算高效的随机涡动力学公式构建的损失函数来近似涡量场。该神经涡量估计器随后与传统数值偏微分方程求解器相结合——可视为网络的最终隐式线性层——通过求解泊松方程来计算速度场。与标准DRVM及其他基于神经网络的数值算法相比，本方法的主要优势在于能够严格保证物理性质，如不可压缩条件或无滑移边界条件，这些性质在其他方法中可能难以保证。通过若干数值实验，验证了我们算法的逼近能力及其融合实测数据的能力。