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.

翻译：我们扩展并分析了针对三维Navier-Stokes方程的深度神经网络多重网格求解器（DNN-MG）。该方法的核心思想是利用深度神经网络获取的细尺度信息，增强粗网格上的有限元模拟。神经网络在网格单元的小型局部块上运行。这种局部方法被证明非常高效，因为网络可以保持（相对）较小，并且可以并行应用于所有网格块。然而，局部方法的主要优势在于其固有的泛化能力。由于网络仅处理小区域的数据，它从未“看见”全局问题，因此不会学习到错误的偏差。我们以有限元方法与深度神经网络之间的交互为重点描述了该方法。此外，我们通过数值示例展示了这种混合方法的卓越效率，使我们能够使用粗网格实现非常高的精度，从而将计算时间减少数个数量级。