Periodic orbit representation of turbulence with automatic differentiation and autoencoders

The dynamical systems view of turbulence promises to connect individual recurrent dynamical processes to statistical properties of the flow. Despite enormous progress in the last few decades, there are still two outstanding challenges in the field: (i) finding and converging exact unstable solutions at high Reynolds numbers ($Re$) and (ii) computing the single set of weights to reconstruct all statistics of the flow with these solutions. In this talk we present new methods to overcome both of these limitations, using two dimensional Kolmogorov flow as a testing ground. We first introduce a new approach to search for periodic orbits via gradient-based optimisation of a scalar loss function, which is effective at high $Re$ and crucially does not require a near recurrence to occur. This approach yields hundreds of solutions where past methods have found only a handful of low-energy states. We then consider the problem of dynamically labelling a turbulent trajectory according to the nearest exact unstable solution in state space using a deep convolutional neural network, in order to compute weights for statistics. The impact of increasing $Re$ on the character of the solutions required for robust statistical reconstructions will also be discussed.