  Traditional reduced order modeling techniques such as the reduced basis (RB)
method (relying, e.g., on proper orthogonal decomposition (POD)) suffer from
severe limitations when dealing with nonlinear time-dependent parametrized
PDEs, because of the fundamental assumption of linear superimposition of modes
they are based on. For this reason, in the case of problems featuring coherent
structures that propagate over time such as transport, wave, or
convection-dominated phenomena, the RB method usually yields inefficient
reduced order models (ROMs) if one aims at obtaining reduced order
approximations sufficiently accurate compared to the high-fidelity, full order
model (FOM) solution. To overcome these limitations, in this work, we propose a
new nonlinear approach to set reduced order models by exploiting deep learning
(DL) algorithms. In the resulting nonlinear ROM, which we refer to as DL-ROM,
both the nonlinear trial manifold (corresponding to the set of basis functions
in a linear ROM) as well as the nonlinear reduced dynamics (corresponding to
the projection stage in a linear ROM) are learned in a non-intrusive way by
relying on DL algorithms; the latter are trained on a set of FOM solutions
obtained for different parameter values. In this paper, we show how to
construct a DL-ROM for both linear and nonlinear time-dependent parametrized
PDEs; moreover, we assess its accuracy on test cases featuring different
parametrized PDE problems. Numerical results indicate that DL-ROMs whose
dimension is equal to the intrinsic dimensionality of the PDE solutions
manifold are able to approximate the solution of parametrized PDEs in
situations where a huge number of POD modes would be necessary to achieve the
same degree of accuracy.
