Reducing data resolution for better super-resolution: Reconstructing turbulent flows from noisy observation

TL;DR

This paper introduces a super-resolution method that reconstructs turbulent flows from noisy data by averaging observations over a coarse grid and employing a dynamic observer, with theoretical and numerical validation showing improved accuracy with coarser resolutions.

Contribution

The paper presents a novel super-resolution approach that combines noise reduction via coarse averaging with dynamic flow reconstruction, supported by theoretical analysis and numerical experiments.

Findings

The SR observer converges exponentially fast to the true flow.

Increasing the spatial averaging scale can reduce deviation from the true flow.

There exists a critical averaging scale for optimal flow reconstruction.

Abstract

A super-resolution (SR) method for the reconstruction of Navier-Stokes (NS) flows from noisy observations is presented. In the SR method, first the observation data is averaged over a coarse grid to reduce the noise at the expense of losing resolution and, then, a dynamic observer is employed to reconstruct the flow field by reversing back the lost information. We provide a theoretical analysis, which indicates a chaos synchronization of the SR observer with the reference NS flow. It is shown that, even with noisy observations, the SR observer converges toward the reference NS flow exponentially fast, and the deviation of the observer from the reference system is bounded. Counter-intuitively, our theoretical analysis shows that the deviation can be reduced by increasing the lengthscale of the spatial average, i.e., making the resolution coarser. The theoretical analysis is confirmed by numerical experiments of two-dimensional NS flows. The numerical experiments suggest that there is a critical lengthscale for the spatial average, below which making the resolution coarser improves the reconstruction.