New computational methods for the dynamical systems view of turbulence

Over the last three decades ideas from dynamical systems theory have significantly advanced our understanding of transitional and weakly turbulent shear flows. In this perspective, the evolution of a turbulent flow is considered as a trajectory in a very high-dimensional dynamical system `pinballing' between unstable exact coherent states (ECS). Applying these ideas to turbulent flows at high Re has the potential to advance our understanding of the role of individual dynamical processes in producing the well-known statistical results (e.g. the cascade) and brings new opportunities for modelling and control. However, progress has stalled due to both an inability to identify guesses for candidate ECS and the poor performance of the Newton-Raphson methods used for convergence. In this talk I will discuss new approaches to both of these problems built on ideas from machine learning, using two-dimensional Kolmogorov flow as an example. First, I will show how deep convolutional autoencoders can be employed to learn low-dimensional representations of the flow which are closely related to ECS. These latent representations form a robust observable with which to measure near recurrences on turbulent orbits, leading to the discovery of an order of magnitude more periodic orbits than standard methods, including a large number of new solutions associated with intermittent, high-dissipation bursts. I will then describe how the requirements for a near recurrence can be removed altogether using a fully-differentiable flow solver (Kochkov et al, Proc. Nat. Acad. Sci. 118, 2021), where periodic orbits with specific properties can be sought via gradient descent on an appropriate loss function. This new method yields large numbers of periodic orbits at high Re, where past methods have found only a handful of structures. Time permitting, I will also discuss data-driven methods for weighting the collection of ECS to estimate flow statistics in an approach akin to periodic orbit theory.