  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.
