However, the solution of parametrized, time-dependent systems of partial differential equations (PDEs) by means of full order models (FOMs) - such as the finite element method - may clash with time and computational budget restrictions. Moreover, using FOMs to explore different scenarios with varying initial conditions and parameter combinations might be a computationally prohibitive task, or even infeasible in several practical applications. Differently from the problem of estimating output quantities of interest that depend on the solution of the differential problem, the computation of the whole solution field is intrinsically high-dimensional, with additional difficulties related to the nonlinear and time-dependent nature of the problem. All these reasons drive the search of efficient, but accurate, reduced order models (ROMs). Among these, the reduced basis method [56, 33, 4] is a very well-known approach, exploiting, e.g., proper orthogonal decomposition (POD) to build a reduced space, either global or local [1, 55], to approximate the solution of the problem. However, despite their accuracy and mathematical, these techniques are in general intrusive [25]. Among machine and deep learning techniques widely used to build surrogate models or emulators to the solution of parametrized, nonlinear, time-dependent system of PDEs, autoencoder (AE) neural networks [27] have recently become a popular strategy because they allow to non-intrusively reduce dimensionality and unveil latent features directly from data streams, without accessing the FOM operators [26, 43, 48, 37]. Their success is due to the expressiveness capacity of neural networks [11, 44, 34], which enables outstanding performances in nonlinear compression and great flexibility in identifying coordinate transformations [46].

Starting from fullorder representations, we apply deep learning techniques to generate accurate, efficient and real-time reduced order models to be used as virtual twin for the simulation and optimization of higher-level complex systems. We extensively test the reliability of the proposed procedures on micromirrors, arches and gyroscopes, also displaying intricate dynamical evolutions like internal resonances. In particular, we discuss the accuracy of the deep learning technique and its ability to replicate and converge to the invariant manifolds predicted using the recently developed direct parametrization approach that allows extracting the nonlinear normal modes of large finite element models. Finally, by addressing an electromechanical gyroscope, we show that the non-intrusive deep learning approach generalizes easily to complex multiphysics problems.