  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.
