The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficiently learn quantities of interest of solutions to a parametric PDE. The central component in our approach is an efficient neural-network-weighted Minimal-Residual formulation, which, after training, provides Galerkin-based approximations in standard discrete spaces that have accurate quantities of interest, regardless of the coarseness of the discrete space.

翻译：参数化偏微分方程的高效近似在科学和工程中具有重要意义。本文展示了如何训练Galerkin离散格式，以高效学习参数化PDE解中的感兴趣量。我们方法的核心是一种高效的神经网络加权最小残差公式，该公式经过训练后，能在标准离散空间中提供基于Galerkin的近似解，无论离散空间粗糙度如何，均可获得精确的感兴趣量。