  Despite the apparent complexity of turbulent flow, identifying a simpler
description of the underlying dynamical system remains a fundamental challenge.
Capturing how the turbulent flow meanders amongst unstable states (simple
invariant solutions) in phase space, as envisaged by Hopf in 1948, using some
efficient representation offers the best hope of doing this, despite the
inherent difficulty in identifying these states. Here, we make a significant
step towards this goal by demonstrating that deep convolutional autoencoders
can identify low-dimensional representations of two-dimensional turbulence
which are closely associated with the simple invariant solutions characterizing
the turbulent attractor. To establish this, we develop latent Fourier analysis
that decomposes the flow embedding into a set of orthogonal latent Fourier
modes which decode into physically meaningful patterns resembling simple
invariant solutions. The utility of this approach is highlighted by analysing
turbulent Kolmogorov flow (flow on a 2D torus forced at large scale) at $Re=40$
where, in between intermittent bursts, the flow resides in the neighbourhood of
an unstable state and is very low dimensional. Projections onto individual
latent Fourier wavenumbers reveal the simple invariant solutions organising
both the quiescent and bursting dynamics in a systematic way inaccessible to
previous approaches.
