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

TL;DR

This paper introduces a neural network-based super-resolution approach within the variational multiscale framework to accurately model unresolved physics in fluid dynamics, demonstrating improved accuracy and generalization across different flow problems.

Contribution

The paper presents a novel neural network architecture, VSRNN, for data-driven closure modeling in VMS formulations, enabling super-resolution of unresolved scales in fluid simulations.

Findings

Improved accuracy in convection-diffusion, advection, and turbulent flow simulations.

Effective generalization to new initial conditions and Reynolds numbers.

Enhanced closure modeling for continuous and discontinuous Galerkin methods.

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 sub-scales to learning the projected element Green's function basis coefficients. For the purpose of this approximation, a special neural-network structure - the variational super-resolution N-N (VSRNN) - is proposed. 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 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 problem, linear advection and turbulent channel flow. Finally, we demonstrate that - in the investigated examples - 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.