  Developing accurate, efficient, and robust closure models is essential in the
construction of reduced order models (ROMs) for realistic nonlinear systems,
which generally require drastic ROM mode truncations. We propose a deep
residual neural network (ResNet) closure learning framework for ROMs of
nonlinear systems. The novel ResNet-ROM framework consists of two steps: (i) In
the first step, we use ROM projection to filter the given nonlinear PDE and
construct a filtered ROM. This filtered ROM is low-dimensional, but is not
closed (because of the PDE nonlinearity). (ii) In the second step, we use
ResNet to close the filtered ROM, i.e., to model the interaction between the
resolved and unresolved ROM modes. We emphasize that in the new ResNet-ROM
framework, data is used only to complement classical physical modeling (i.e.,
only in the closure modeling component), not to completely replace it. We also
note that the new ResNet-ROM is built on general ideas of spatial filtering and
deep learning and is independent of (restrictive) phenomenological arguments,
e.g., of eddy viscosity type. The numerical experiments for the 1D Burgers
equation show that the ResNet-ROM is significantly more accurate than the
standard projection ROM. The new ResNet-ROM is also more accurate and
significantly more efficient than other modern ROM closure models.
