  We extend and analyze the deep neural network multigrid solver (DNN-MG) for
the Navier-Stokes equations in three dimensions. The idea of the method is to
augment a finite element simulation on coarse grids with fine scale information
obtained using deep neural networks.
  The neural network operates locally on small patches of grid elements. The
local approach proves to be highly efficient, since the network can be kept
(relatively) small and since it can be applied in parallel on all grid patches.
However, the main advantage of the local approach is the inherent
generalizability of the method. Since the network only processes data of small
sub-areas, it never ``sees'' the global problem and thus does not learn false
biases.
  We describe the method with a focus on the interplay between the finite
element method and deep neural networks. Further, we demonstrate with numerical
examples the excellent efficiency of the hybrid approach, which allows us to
achieve very high accuracy with a coarse grid and thus reduce the computation
time by orders of magnitude.
