In this paper we introduce a novel Neural Networks-based approach for approximating solutions to the (2D) incompressible Navier--Stokes equations. Our algorithm uses a Physics-informed Neural Network, 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 for the Poisson equation to compute the velocity field. The main advantage of our method compared to standard Physics-informed Neural Networks is that it strictly enforces physical properties, such as incompressibility or boundary conditions, which might otherwise be hard to guarantee with purely Neural Networks-based approaches.

翻译：本文提出了一种新颖的基于神经网络的方法，用于近似求解二维不可压缩Navier-Stokes方程的解。我们的算法采用物理信息神经网络，该网络基于随机涡动力学的计算高效公式构建损失函数来近似涡量。随后，该神经涡量估计器与传统的泊松方程数值PDE求解器相结合，以计算速度场。与标准物理信息神经网络相比，本方法的主要优势在于其严格保证了物理性质（如不可压缩性或边界条件），而这些性质在纯基于神经网络的方法中可能难以保证。