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We extend and analyze the deep neural network multigrid solver (DNN-MG) forthe Navier-Stokes equations in three dimensions. The idea of the method is toaugment a finite element simulation on coarse grids with fine scale informationobtained using deep neural networks.  The neural network operates locally on small patches of grid elements. Thelocal 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 inherentgeneralizability of the method. Since the network only processes data of smallsub-areas, it never ``sees'' the global problem and thus does not learn falsebiases.  We describe the method with a focus on the interplay between the finiteelement method and deep neural networks. Further, we demonstrate with numericalexamples the excellent efficiency of the hybrid approach, which allows us toachieve very high accuracy with a coarse grid and thus reduce the computationtime by orders of magnitude.

DNN-MG: A Hybrid Neural Network/Finite Element Method with Applications to 3D Simulations of the Navier-Stokes Equations