We analyze a hybrid method that enriches coarse grid finite element solutions with fine scale fluctuations obtained from a neural network. The idea stems from the Deep Neural Network Multigrid Solver (DNN-MG), (Margenberg et al., J Comput Phys 460:110983, 2022; A neural network multigrid solver for the Navier-Stokes equations) which embeds a neural network into a multigrid hierarchy by solving coarse grid levels directly and predicting the corrections on fine grid levels locally (e.g. on small patches that consist of several cells) by a neural network. Such local designs are quite appealing, as they allow a very good generalizability. In this work, we formalize the method and describe main components of the a-priori error analysis. Moreover, we numerically investigate how the size of training set affects the solution quality.

翻译：我们分析了一种混合方法，该方法利用神经网络获取的细尺度波动来丰富粗网格有限元解。这一思想源于深度神经网络多重网格求解器（DNN-MG）（Margenberg 等，《计算物理学杂志》460:110983, 2022；《用于纳维-斯托克斯方程的神经网络多重网格求解器》），该方法通过直接求解粗网格层级，并在细网格层级上局部（例如，由若干单元组成的小块区域）使用神经网络预测修正，从而将神经网络嵌入到多重网格层级结构中。这种局部设计颇具吸引力，因为它具有良好的泛化能力。在本文中，我们对该方法进行了形式化描述，并阐述了先验误差分析的主要组成部分。此外，我们通过数值实验研究了训练集的大小对求解质量的影响。