Exact coherent structures in two-dimensional turbulence identified with convolutional autoencoders

Abstract

Convolutional autoencoders are used to deconstruct the changing dynamics of two-dimensional Kolmogorov flow as $Re$ is increased from weakly chaotic flow at $Re=40$ to a chaotic state dominated by a domain-filling vortex pair at $Re=400$ . ‘Latent Fourier analysis’ (Page et al., Phys. Rev. Fluids6, 2021, p. 034402) reveals a detached class of bursting dynamics at $Re=40$ which merge with the low-dissipation dynamics as $Re$ is increased to $100$ and provides an efficient representation within which to find unstable periodic orbits (UPOs) using recurrent flow analysis. Focusing on initial guesses with energy in higher latent Fourier wavenumbers allows a significant number of high-dissipation-rate UPOs associated with the bursting events to be found for the first time. At $Re=400$ , the UPOs discovered at lower $Re$ move away from the attractor, and an entirely different embedding structure is formed within the network devoid of small-scale vortices. Here latent Fourier projections identify an associated ‘large-scale’ UPO which we believe to be a finite- $Re$ continuation of a solution to the Euler equations.