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 of finite element simulations on coarse grids with fine scale information obtained using deep neural networks. This 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 good generalizability of the method. Since the network is only ever trained on small sub-areas, it never ``sees'' the global problem and thus does not learn a false bias. We describe the method with a focus on the interplay between 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 accuracies on coarse grids and thus reduce the computation time by orders of magnitude.

翻译：我们扩展并分析了用于三维Navier-Stokes方程的深度神经网络多重网格求解器（DNN-MG）。该方法的核心思想在于，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。该网络以局部方式作用于网格单元的小型补丁上。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补丁。然而，局部方法的主要优势在于其固有的良好泛化能力。由于网络仅在小区域上进行训练，它永远不会“看到”全局问题，因此不会学习到错误的偏差。我们重点描述了有限元方法与深度神经网络之间的交互关系。此外，通过数值算例，我们展示了这种混合方法的卓越效率，使我们能够在粗网格上实现非常高的精度，从而将计算时间降低数个数量级。