  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.
