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Despite the apparent complexity of turbulent flow, identifying a simplerdescription of the underlying dynamical system remains a fundamental challenge.Capturing how the turbulent flow meanders amongst unstable states (simpleinvariant solutions) in phase space, as envisaged by Hopf in 1948, using someefficient representation offers the best hope of doing this, despite theinherent difficulty in identifying these states. Here, we make a significantstep towards this goal by demonstrating that deep convolutional autoencoderscan identify low-dimensional representations of two-dimensional turbulencewhich are closely associated with the simple invariant solutions characterizingthe turbulent attractor. To establish this, we develop latent Fourier analysisthat decomposes the flow embedding into a set of orthogonal latent Fouriermodes which decode into physically meaningful patterns resembling simpleinvariant solutions. The utility of this approach is highlighted by analysingturbulent 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 ofan unstable state and is very low dimensional. Projections onto individuallatent Fourier wavenumbers reveal the simple invariant solutions organisingboth the quiescent and bursting dynamics in a systematic way inaccessible toprevious approaches.

Revealing the state space of turbulence using machine learning