Deep-HyROMnet: A Deep Learning-Based Operator Approximation for Hyper-Reduction of Nonlinear Parametrized PDEs

Abstract To speed-up the solution of parametrized differential problems, reduced order models (ROMs) have been developed over years, including projection-based ROMs such as reduced-basis (RB) method, deep learning-based ROMs, well surrogate obtained through machine learning techniques. Thanks to its physics-based structure, ensured by use a Galerkin projection full model (FOM) onto linear low-dimensional subspace, Galerkin-RB method yields approximations that fulfill problem at hand. However, make assembling ROM independent FOM dimension, intrusive and expensive hyper-reduction techniques, discrete empirical interpolation (DEIM), are usually required, thus making this strategy less feasible for problems characterized (high-order polynomial or nonpolynomial) nonlinearities. overcome bottleneck, we propose novel nonlinear operators using neural networks (DNNs). The resulting hyper-reduced enhanced DNNs, which refer Deep-HyROMnet, is then model, still relying on RB approach, however employing DNN architecture approximate residual vectors Jacobian matrices once has performed. Numerical results dealing with fast simulations in structural mechanics show Deep-HyROMnets orders magnitude faster than POD-Galerkin-DEIM ensuring same level accuracy.