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

TL;DR

This paper introduces DNN-MG, a hybrid solver combining neural networks and finite element methods for 3D Navier-Stokes simulations, achieving high accuracy and efficiency by local processing and parallelization.

Contribution

The paper presents a novel hybrid neural network and finite element method that enhances 3D Navier-Stokes simulations with improved accuracy and computational efficiency.

Findings

Achieves high accuracy with coarse grids

Reduces computation time by orders of magnitude

Demonstrates excellent efficiency in numerical examples

Abstract

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.