  We present the deep neural network multigrid solver (DNN-MG) that we develop
for the instationary Navier-Stokes equations. DNN-MG improves computational
efficiency using a judicious combination of a geometric multigrid solver and a
recurrent neural network with memory. DNN-MG uses the multi-grid method to
classically solve on coarse levels while the neural network corrects
interpolated solutions on fine ones, thus avoiding the increasingly expensive
computations that would have to be performed there. This results in a reduction
in computation time through DNN-MG's highly compact neural network. The
compactness results from its design for local patches and the available coarse
multigrid solutions that provides a "guide" for the corrections. A compact
neural network with a small number of parameters also reduces training time and
data. Furthermore, the network's locality facilitates generalizability and
allows one to use DNN-MG trained on one mesh domain also on different ones. We
demonstrate the efficacy of DNN-MG for variations of the 2D laminar flow around
an obstacle. For these, our method significantly improves the solutions as well
as lift and drag functionals while requiring only about half the computation
time of a full multigrid solution. We also show that DNN-MG trained for the
configuration with one obstacle can be generalized to other time dependent
problems that can be solved efficiently using a geometric multigrid method.
