Score-based stochastic reduced-order models of barotropic quasi-geostrophic turbulence

Abstract

In many turbulent flows, the quantities of interest are coarse-grained variables, either because resolving the fine scales is computationally prohibitive or because only coarse observations are available; in both cases, data-driven reduced-order models that evolve only these variables are required. Even when the governing equations are known, their projection onto the coarse variables is not closed, and the central difficulty lies in representing the feedback of the unresolved scales so that the reduced model reproduces the statistical and dynamical properties of the projected high-resolution system. We address this problem with a score-based framework, constructing a reduced-order model of quasi-geostrophic turbulence whose stationary distribution coincides by construction with the distribution learned from data and whose dynamics are calibrated to reproduce the finite-time correlations of the energies in individual wavenumber shells. The resulting model reproduces the stationary statistics of the resolved flow with high accuracy, and its calibration is computationally inexpensive because it requires no forward integration of the model.