  Model order reduction through the POD-Galerkin method can lead to dramatic
gains in terms of computational efficiency in solving physical problems.
However, the applicability of the method to non linear high-dimensional
dynamical systems such as the Navier-Stokes equations has been shown to be
limited, producing inaccurate and sometimes unstable models. This paper
proposes a deep learning based closure modeling approach for classical
POD-Galerkin reduced order models (ROM). The proposed approach is theoretically
grounded, using neural networks to approximate well studied operators. In
contrast with most previous works, the present CD-ROM approach is based on an
interpretable continuous memory formulation, derived from simple hypotheses on
the behavior of partially observed dynamical systems. The final corrected
models can hence be simulated using most classical time stepping schemes. The
capabilities of the CD-ROM approach are demonstrated on two classical examples
from Computational Fluid Dynamics, as well as a parametric case, the
Kuramoto-Sivashinsky equation.
