Variational multiscale super-resolution: A data-driven approach for reconstruction and predictive modeling of unresolved physics

Abstract

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. In linear problems, our formulation reduces the problem of learning the subscales to learning the projected element Green's function basis coefficients. For this approximation, a special neural-network structure—the variational super-resolution N-N (VSRNN)—is proposed. The VSRNN constructs a super-resolution 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 nondimensional features obtained by normalizing the input coarse-scale and output subscale basis coefficients, the VSRNN provides a general framework for the discovery of closures for different Galerkin discretizations. By training it on a sequence of projected data and using the subscale to compute the continuous Galerkin subgrid terms, and the super-resolved state to compute the discontinuous Galerkin fluxes, we improve the optimality and the accuracy of these methods for the convection-diffusion and linear advection problems. Finally, we demonstrate that the VSRNN allows generalization to out-of-sample initial conditions and nondimensional numbers.