Dynamics-based machine learning of transitions in shear flows

Several approaches have been proposed to deriving reduced-order models for exact coherent states (ECSs) and transitions among them in canonical shear flows, such as the plane Couette flow. One such approach uses truncated modal projections of the Navier-Stokes equations. This technique may lead to reasonably accurate models but fails to provide a priori estimates for the necessary number of modes to be included and the magnitude of the truncation error. Alternative approaches include the Dynamic Mode Decomposition (DMD and Koopman decomposition, which fit linear dynamical systems to a set of observables. By their linearity, however, these approaches cannot produce reduced models that capture fundamentally nonlinear phenomena, such as coexisting ECSs and transitions among them.

The theory of spectral submanifolds (SSMs) offers an alternative for reduced-order modeling of phenomena involving multiple, coexisting ECSs. SSMs are very low dimensional attracting invariant manifolds that emanate from stationary states and have the potential to connect those states to other coexisting stationary states. Recent advances have enabled the identification of SSMs purely from numerical or experimental data. Here, we apply these results to construct SSMs of simple ECSs in plane Couette flow. We show that the energy input and output rate flow can be used to parametrize the most important SSMs connecting coexisting ECSs. By restricting the dynamics to these SSMs, we obtain accurate reduced-order models that capture all asymptotic states of the full Navier-Stokes equations. A similar approach is expected to apply to pipe flow and explain aspects of the subcritical transition to turbulence.