Data-driven Koopman operator predictions of turbulent dynamics in models of shear flows

The Koopman operator enables the analysis of nonlinear dynamical systems through a linear perspective by describing time evolution in the infinite-dimensional space of observables. Here this formalism is applied to shear flows, specifically the minimal flow unit plane Couette flow and a nine-dimensional model of turbulent shear flow between infinite parallel free-slip walls under a sinusoidal body force (Moehlis et al., New J. Phys. 6, 56 (2004)). We identify a finite set of observables and approximate the Koopman operator using neural networks, following the method developed by Constante-Amores et al., Chaos, 34(4), 043119 (2024). Then the time evolution is determined with a method here denoted as "Projected Koopman Dynamics". Using a high-dimensional approximate Koopman operator directly for the plane Couette system is computationally infeasible due to the high state space dimension of direct numerical simulations (DNS). However, the long-term dynamics of the dissipative Navier-Stokes equations are expected to live on a lower-dimensional invariant manifold. Motivated by this fact, we perform dimension reduction on the DNS data with a variant of the IRMAE-WD (implicit rank-minimizing autoencoder-weight decay) autoencoder (Zeng et al., MLST 5(2), 025053 (2024)) before applying the Koopman approach. We compare the performance of data-driven state space (neural ordinary differential equation) and Projected Koopman Dynamics approaches in manifold coordinates for plane Couette flow. Projected Koopman Dynamics effectively captures the chaotic behavior of Couette flow and the nine-dimensional model, excelling in both short-term tracking and long-term statistical analysis, outperforming other frameworks applied to these shear flows.