  We investigate scaling and efficiency of the deep neural network multigrid
method (DNN-MG).
  DNN-MG is a novel neural network-based technique for the simulation of the
Navier-Stokes equations that combines an adaptive geometric multigrid solver,
i.e. a highly efficient
  classical solution scheme, with a recurrent neural network with memory.
  The neural network replaces in DNN-MG one or multiple finest multigrid layers
and provides a correction for the classical solve in the next time step.
  This leads to little degradation in the solution quality while substantially
reducing the overall computational costs.
  At the same time, the use of the multigrid solver at the coarse scales allows
for a compact network that is easy to train, generalizes well, and allows for
the incorporation of physical constraints.
  Previous work on DNN-MG focused on the overall scheme and how to enforce
divergence freedom in the solution.
  In this work, we investigate how the network size affects training and
solution quality and the overall runtime of the computations.
  Our results demonstrate that larger networks are able to capture the
  flow behavior better while requiring only little additional training time.
  At runtime, the use of the neural network correction can even reduce the
computation time compared to a classical multigrid simulation through a faster
convergence of the nonlinear solve that is required at every time step.
