To solve high-dimensional parameter-dependent partial differential equations (pPDEs), a neural network architecture is presented. It is constructed to map parameters of the model data to corresponding finite element solutions. To improve training efficiency and to enable control of the approximation error, the network mimics an adaptive finite element method (AFEM). It outputs a coarse grid solution and a series of corrections as produced in an AFEM, allowing a tracking of the error decay over successive layers of the network. The observed errors are measured by a reliable residual based a posteriori error estimator, enabling the reduction to only few parameters for the approximation in the output of the network. This leads to a problem adapted representation of the solution on locally refined grids. Furthermore, each solution of the AFEM is discretized in a hierarchical basis. For the architecture, convolutional neural networks (CNNs) are chosen. The hierarchical basis then allows to handle sparse images for finely discretized meshes. Additionally, as corrections on finer levels decrease in amplitude, i.e., importance for the overall approximation, the accuracy of the network approximation is allowed to decrease successively. This can either be incorporated in the number of generated high fidelity samples used for training or the size of the network components responsible for the fine grid outputs. The architecture is described and preliminary numerical examples are presented.

翻译：为求解高维参数相关的偏微分方程（pPDEs），提出了一种神经网络架构。该架构将模型数据参数映射至对应的有限元解。为提升训练效率并实现近似误差的可控性，该网络模拟了自适应有限元方法（AFEM）。网络输出粗网格解及一系列修正量（与AFEM过程一致），从而可追踪误差在网络各层的衰减过程。观测误差通过基于残差的后验误差估计器进行可靠度量，这使得网络输出中的近似仅需少量参数即可实现。由此可得到局部加密网格上的问题自适应解。此外，AFEM的每个解均采用层次基离散化。在架构层面，选用卷积神经网络（CNN）。层次基的实现使得精细离散化网格的稀疏图像处理成为可能。同时，由于精细层级上的修正量振幅（即对整体近似的重要性）逐步降低，网络近似的精度也可逐级下降。这一特性可通过调整训练用高保真样本数量，或控制负责精细网格输出的网络组件规模来实现。本文阐述了该架构的设计思路，并给出了初步数值算例。