  Deep learning-based reduced order models (DL-ROMs) have been recently
proposed to overcome common limitations shared by conventional ROMs - built,
e.g., exclusively through proper orthogonal decomposition (POD) - when applied
to nonlinear time-dependent parametrized PDEs. In particular, POD-DL-ROMs can
achieve extreme efficiency in the training stage and faster than real-time
performances at testing, thanks to a prior dimensionality reduction through POD
and a DL-based prediction framework. Nonetheless, they share with conventional
ROMs poor performances regarding time extrapolation tasks. This work aims at
taking a further step towards the use of DL algorithms for the efficient
numerical approximation of parametrized PDEs by introducing the $\mu
t$-POD-LSTM-ROM framework. This novel technique extends the POD-DL-ROM
framework by adding a two-fold architecture taking advantage of long short-term
memory (LSTM) cells, ultimately allowing long-term prediction of complex
systems' evolution, with respect to the training window, for unseen input
parameter values. Numerical results show that this recurrent architecture
enables the extrapolation for time windows up to 15 times larger than the
training time domain, and achieves better testing time performances with
respect to the already lightning-fast POD-DL-ROMs.
