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Searching for periodic orbits in turbulence with convolutional neural networks

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Unstable periodic orbits (UPOs) are the building blocks of chaotic attractors and possibly turbulent attractors too. The current state-of-the-art for finding UPOs in a turbulent flow begins with a search for `near recurrences’ in a DNS time series, measured as local minima in an $l_2$-norm between snapshots of the flow. The approach is crude and struggles to identify UPOs which are visited only fleetingly or which may be spatially localised. In this work we explore the use of convolutional neural networks (CNNs) as a means of performing a dimensionality reduction that respects the existence of UPOs and which can then be applied as a tool for efficiently identifying these coherent structures in turbulent data streams. We train a CNN in the form of an autoencoder to reconstruct snapshots of turbulent Kolmogorov flow (body-forced Navier Stokes equations on a 2-torus). The autoencoder reduces the dimensionality of the flow by orders of magnitude while its output is largely indistinguishable from the true turbulence. The network naturally develops an embedding of the continuous translational symmetry in the system, and we exploit this fact to define translation-independent observables of encoded vorticity fields. These observables can be used as a visualisation tool for comparing encoded UPOs, which cluster into distinct families of coherent structures with different dynamic features. The suggestion that the network has learnt a dimensionality reduction that is related to the exact coherent structures is confirmed by performing a recurrent flow analysis on encoded time series using the translation-independent observable. The approach results in the identification of an order of magnitude more UPOs as compared to a standard recurrent flow analysis over the same time interval. We will go on to assess the network’s performance at higher Reynolds numbers, where only a handful of exact coherent structures have been previously identified.

This talk is part of the BPI Seminar Series series.

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