  Deep learning-based reduced order models (DL-ROMs) have been recently
proposed to overcome common limitations shared by conventional reduced order
models (ROMs) - built, e.g., through proper orthogonal decomposition (POD) -
when applied to nonlinear time-dependent parametrized partial differential
equations (PDEs). These might be related to (i) the need to deal with
projections onto high dimensional linear approximating trial manifolds, (ii)
expensive hyper-reduction strategies, or (iii) the intrinsic difficulty to
handle physical complexity with a linear superimposition of modes. All these
aspects are avoided when employing DL-ROMs, which learn in a non-intrusive way
both the nonlinear trial manifold and the reduced dynamics, by relying on deep
(e.g., feedforward, convolutional, autoencoder) neural networks. Although
extremely efficient at testing time, when evaluating the PDE solution for any
new testing-parameter instance, DL-ROMs require an expensive training stage,
because of the extremely large number of network parameters to be estimated. In
this paper we propose a possible way to avoid an expensive training stage of
DL-ROMs, by (i) performing a prior dimensionality reduction through POD, and
(ii) relying on a multi-fidelity pretraining stage, where different physical
models can be efficiently combined. The proposed POD-DL-ROM is tested on
several (both scalar and vector, linear and nonlinear) time-dependent
parametrized PDEs (such as, e.g., linear advection-diffusion-reaction,
nonlinear diffusion-reaction, nonlinear elastodynamics, and Navier-Stokes
equations) to show the generality of this approach and its remarkable
computational savings.
