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

Abstract

AbstractThe 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.