Variational Multi-scale Super-resolution : A data-driven approach for reconstruction and predictive modeling of unresolved physics.

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.