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

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. The highly accurate embeddings allow us to visualise the evolving structure of state space and are interpretable using `latent Fourier analysis' (Page {\em et. al.}, \emph{Phys. Rev. Fluids} \textbf{6}, 2021). Individual latent Fourier modes decode into vortical structures with a streamwise lengthscale controlled by the latent wavenumber, l, with only a small number l≲8 required to accurately represent the flow. Latent Fourier projections reveal a detached class of bursting events at Re=40 which merge with the low-dissipation dynamics as Re is increased to 100. We use doubly- (l=2) or triply- (l=3) periodic latent Fourier modes to generate guesses for UPOs (unstable periodic orbits) associated with high-dissipation events. While the doubly-periodic UPOs are representative of the high-dissipation dynamics at Re=40, the same class of UPOs move away from the attractor at Re=100 -- where the associated bursting events typically involve larger-scale (l=1) structure too. At Re=400 an entirely different embedding structure is formed within the network in which no distinct representations of small-scale vortices are observed; instead the network embeds all snapshots based around a large-scale template for the condensate. We use latent Fourier projections to find an associated `large-scale' UPO which we believe to be a finite-Re continuation of a solution to the Euler equations.