  Projection-based reduced order models (PROMs) have shown promise in
representing the behavior of multiscale systems using a small set of
generalized (or latent) variables. Despite their success, PROMs can be
susceptible to inaccuracies, even instabilities, due to the improper accounting
of the interaction between the resolved and unresolved scales of the multiscale
system (known as the closure problem). In the current work, we interpret
closure as a multifidelity problem and use a multifidelity deep operator
network (DeepONet) framework to address it. In addition, to enhance the
stability and accuracy of the multifidelity-based closure, we employ the
recently developed "in-the-loop" training approach from the literature on
coupling physics and machine learning models. The resulting approach is tested
on shock advection for the one-dimensional viscous Burgers equation and vortex
merging using the two-dimensional Navier-Stokes equations. The numerical
experiments show significant improvement of the predictive ability of the
closure-corrected PROM over the un-corrected one both in the interpolative and
the extrapolative regimes.
