Title: A deep learning approach to solve parametric PDEs in terms of a quantity of interest
Abstract: In this talk, we will present a Minimal-Residual (MinRes) framework, where we can tune a finite element scheme using neural networks to accurately solve parametric families of PDEs in terms of a quantity of interest (QoI). The main idea is to combine neural networks with Finite Element (FE) schemes on coarse meshes to approximate (at high precision) QoIs of parametric PDEs over low-dimensional FE spaces. The NN will deliver a weighted inner product for the test space, where the training is performed on a supervised machine-learning setting and inner-product weights are tuned against available training data. With the extra degrees of freedom given by the weights, we can explore different FE schemes and find the one that approximates the QoI for any PDE within the parametric family.