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