  In the field of fluid numerical analysis, there has been a long-standing
problem: lacking of a rigorous mathematical tool to map from a continuous flow
field to discrete vortex particles, hurdling the Lagrangian particles from
inheriting the high resolution of a large-scale Eulerian solver. To tackle this
challenge, we propose a novel learning-based framework, the Neural Vortex
Method (NVM), which builds a neural-network description of the Lagrangian
vortex structures and their interaction dynamics to reconstruct the
high-resolution Eulerian flow field in a physically-precise manner. The key
components of our infrastructure consist of two networks: a vortex
representation network to identify the Lagrangian vortices from a grid-based
velocity field and a vortex interaction network to learn the underlying
governing dynamics of these finite structures. By embedding these two networks
with a vorticity-to-velocity Poisson solver and training its parameters using
the high-fidelity data obtained from high-resolution direct numerical
simulation, we can predict the accurate fluid dynamics on a precision level
that was infeasible for all the previous conventional vortex methods (CVMs). To
the best of our knowledge, our method is the first approach that can utilize
motions of finite particles to learn infinite dimensional dynamic systems. We
demonstrate the efficacy of our method in generating highly accurate prediction
results, with low computational cost, of the leapfrogging vortex rings system,
the turbulence system, and the systems governed by Euler equations with
different external forces.
