Searching for periodic orbits in turbulent flows using machine learning

Our understanding of fluid turbulence has improved through the discovery of large numbers of unstable solutions to the Navier-Stokes equations. A turbulent trajectory bounces between these exact coherent states (ECS), being drawn in and then flung out along their stable and unstable manifolds. This low-dimensional view of turbulence proves difficult to extend beyond moderate Reynolds numbers due to inherent difficulties in identifying the ECS which are visited only fleetingly, while standard dimensionality reduction techniques like POD are unrelated to and mask ECS. Motivated by these observations, we explore whether deep neural networks can be used to identify low-order projections of turbulence that respect the existence of ECS by training a convolutional autoencoder to reconstruct snapshots from turbulent 2D Kolmogorov flows. The encoded representation reduces the number of degrees of freedom by several of orders of magnitude. We then compute the encoded representations of a large number of ECS to see how well separated these solutions are in the encoded space. Finally, we discuss how the approach can be used to extract ECS at high Reynolds number.