A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations

We introduce the concept of data-driven finite element methods. These are finite-element discretizations partial differential equations (PDEs) that resolve quantities interest with striking accuracy, regardless underlying mesh size. The methods obtained within a machine-learning framework during which parameters defining method tuned against available training data. In particular, we use stable parametric Petrov-Galerkin is equivalent to minimal-residual formulation using weighted norm. While trial space standard space, test has in an off-line stage. Finding optimal therefore amounts obtaining goal-oriented discretization completely tailored towards quantity interest. As natural deep learning, artificial neural network define family spaces. Using numerical examples for Laplacian and advection equation one two dimensions, demonstrate superior approximation even on very coarse meshes