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逼近在科学与工程领域具有极其重要的价值。本文展示了如何训练Galerkin离散格式以高效学习参数化偏微分方程解的感兴趣量。该方法的核心是一种高效的神经网络加权最小残差公式，经过训练后，该公式能在标准离散空间中提供基于Galerkin的近似解，且无论离散空间粗糙程度如何，均能获得精确的感兴趣量。