  The simulation of partial differential equations is a central subject of
numerical analysis and an indispensable tool in science, engineering and
related fields. Existing approaches, such as finite elements, provide (highly)
efficient tools but deep neural network-based techniques emerged in the last
few years as an alternative with very promising results. We investigate the
combination of both approaches for the approximation of the Navier-Stokes
equations and to what extent structural properties such as divergence freedom
can and should be respected. Our work is based on DNN-MG, a deep neural network
multigrid technique, that we introduced recently and which uses a neural
network to represent fine grid fluctuations not resolved by a geometric
multigrid finite element solver. Although DNN-MG provides solutions with very
good accuracy and is computationally highly efficient, we noticed that the
neural network-based corrections substantially violate the divergence freedom
of the velocity vector field. In this contribution, we discuss these findings
and analyze three approaches to address the problem: a penalty term to
encourage divergence freedom of the network output; a penalty term for the
corrected velocity field; and a network that learns the stream function, i.e.
the scalar potential of the divergence free velocity vector field and which
hence yields by construction divergence free corrections. Our experimental
results show that the third approach based on the stream function outperforms
the other two and not only improves the divergence freedom but in particular
also the overall fidelity of the simulation.
