Data-Driven Goal-Oriented Finite Element Methods: A~Machine-Learning Minimal-Residual (ML-MRes) Framework

Abstract

We consider the data-driven acceleration of Galerkin-based finite element discretizations for the approximation of partial differential equations (PDEs) [1]. The aim is to obtain approximations on meshes that are very coarse, but nevertheless resolve quantities of interest with striking accuracy.Our work is inspired by the the machine learning framework of Mishra [2], who considered the data-driven acceleration of finite-difference schemes. The essential idea is to optimize a numerical method for a given coarse mesh, by minimizing a loss function consisting of errors with respect to the quantities of interest for obtained training data.