  The coupling of Proper Orthogonal Decomposition (POD) and deep learning-based
ROMs (DL-ROMs) has proved to be a successful strategy to construct
non-intrusive, highly accurate, surrogates for the real time solution of
parametric nonlinear time-dependent PDEs. Inexpensive to evaluate, POD-DL-ROMs
are also relatively fast to train, thanks to their limited complexity. However,
POD-DL-ROMs account for the physical laws governing the problem at hand only
through the training data, that are usually obtained through a full order model
(FOM) relying on a high-fidelity discretization of the underlying equations.
Moreover, the accuracy of POD-DL-ROMs strongly depends on the amount of
available data. In this paper, we consider a major extension of POD-DL-ROMs by
enforcing the fulfillment of the governing physical laws in the training
process -- that is, by making them physics-informed -- to compensate for
possible scarce and/or unavailable data and improve the overall reliability. To
do that, we first complement POD-DL-ROMs with a trunk net architecture,
endowing them with the ability to compute the problem's solution at every point
in the spatial domain, and ultimately enabling a seamless computation of the
physics-based loss by means of the strong continuous formulation. Then, we
introduce an efficient training strategy that limits the notorious
computational burden entailed by a physics-informed training phase. In
particular, we take advantage of the few available data to develop a low-cost
pre-training procedure; then, we fine-tune the architecture in order to further
improve the prediction reliability. Accuracy and efficiency of the resulting
pre-trained physics-informed DL-ROMs (PTPI-DL-ROMs) are then assessed on a set
of test cases ranging from non-affinely parametrized
advection-diffusion-reaction equations, to nonlinear problems like the
Navier-Stokes equations for fluid flows.
