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The variational multiscale (VMS) formulation formally segregates theevolution of the coarse-scales from the fine-scales. VMS modeling requires theapproximation of the impact of the fine scales in terms of the coarse scales.In linear problems, our formulation reduces the problem of learning thesub-scales to learning the projected element Green's function basiscoefficients. For the purpose of this approximation, a special neural-networkstructure - the variational super-resolution N-N (VSRNN) - is proposed. TheVSRNN constructs a super-resolved model of the unresolved scales as a sum ofthe products of individual functions of coarse scales and physics-informedparameters. Combined with a set of locally non-dimensional features obtained bynormalizing the input coarse-scale and output sub-scale basis coefficients, theVSRNN provides a general framework for the discovery of closures for both thecontinuous and the discontinuous Galerkin discretizations. By training thismodel on a sequence of $L_2-$projected data and using the subscale to computethe continuous Galerkin subgrid terms, and the super-resolved state to computethe discontinuous Galerkin fluxes, we improve the optimality and the accuracyof these methods for the convection-diffusion problem, linear advection andturbulent channel flow. Finally, we demonstrate that - in the investigatedexamples - the present model allows generalization to out-of-sample initialconditions and Reynolds numbers. Perspectives are provided on data-drivenclosure modeling, limitations of the present approach, and opportunities forimprovement.

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