Neural vortex method: From finite Lagrangian particles to infinite dimensional Eulerian dynamics

Abstract

In fluid analysis, there has been a long-standing problem: lacking a rigorous mathematical tool to map from a continuous flow field to finite discrete 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). NVM 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 detection network to identify the Lagrangian vortices from a grid-based velocity field and a vortex dynamics network to learn the underlying governing interactions of these finite structures. By embedding these two networks with a vorticity-to-velocity Poisson solver and training its parameters using the fluid data obtained from grid-based numerical simulation, we can predict the accurate fluid dynamics on a precision level that was infeasible for all the previous conventional vortex methods. We demonstrate the efficacy of our method in generating highly accurate prediction results with low computational cost by predicting the evolution of the leapfrogging vortex rings system, the turbulence system, and the systems governed by Navier–Stokes (NS) equations with different external forces. We compare the prediction results made by NVM and the Lagrangian vortex method (LVM) for solving the NS equation in the periodic box and find that the relative error of the predicted velocity using NVM is more than 10 times lower than that of the LVM. Moreover, our method only requires data collected from a very short training window, more than 100 times smaller than the prediction period, which potentially facilitates data acquisition in real systems.