Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional ROMs - built, e.g., through proper orthogonal decomposition (POD) - when applied to nonlinear time-dependent parametrized PDEs. Although extremely efficient at testing time, when evaluating the PDE solution for any new testing-parameter instance, DL-ROMs require an expensive training stage. To avoid this latter, a prior dimensionality reduction through POD, and a multi-fidelity pretraining stage, are introduced, yielding the POD-DL-ROM framework, which allows to solve time-dependent PDEs even faster than in real-time. Equipped with LSTM networks, the resulting POD-LSTM-ROMs better grasp the time evolution of the PDE system, ultimately allowing long-term prediction of complex systems’ evolution, with respect to the training window, for unseen input parameter values.