  The variational multiscale (VMS) formulation formally segregates the
evolution of the coarse-scales from the fine-scales. VMS modeling requires the
approximation of the impact of the fine scales in terms of the coarse scales.
For the purpose of this approximation, this work introduces a VMS framework
with a special neural-network (N-N) structure, which we call the variational
super-resolution N-N (VSRNN). The VSRNN constructs a super-resolved model of
the unresolved scales as a sum of the products of individual functions of
coarse scales and physics-informed parameters. Combined with a set of locally
non-dimensional features obtained by normalizing the input coarse-scale and
output sub-scale basis coefficients, the VSRNN provides a general framework for
the discovery of closures for both the continuous and the discontinuous
Galerkin discretizations. By training this model on a sequence of
$L_2-$projected data and using the super-resolved state to compute the
discontinuous Galerkin fluxes, we improve the optimality and the accuracy of
the method for both the linear advection problem and turbulent channel flow.
Finally, we demonstrate that - in the investigated examples - that the present
model allows generalization to out-of-sample initial conditions and Reynolds
numbers. Perspectives are provided on data-driven closure modeling, limitations
of the present approach, and opportunities for improvement.
