  Several related works have introduced Koopman-based Machine Learning
architectures as a surrogate model for dynamical systems. These architectures
aim to learn non-linear measurements (also known as observables) of the
system's state that evolve by a linear operator and are, therefore, amenable to
model-based linear control techniques. So far, mainly simple systems have been
targeted, and Koopman architectures as reduced-order models for more complex
dynamics have not been fully explored. Hence, we use a Koopman-inspired
architecture called the Linear Recurrent Autoencoder Network (LRAN) for
learning reduced-order dynamics in convection flows of a Rayleigh B\'enard
Convection (RBC) system at different amounts of turbulence. The data is
obtained from direct numerical simulations of the RBC system. A traditional
fluid dynamics method, the Kernel Dynamic Mode Decomposition (KDMD), is used to
compare the LRAN. For both methods, we performed hyperparameter sweeps to
identify optimal settings. We used a Normalized Sum of Square Error measure for
the quantitative evaluation of the models, and we also studied the model
predictions qualitatively. We obtained more accurate predictions with the LRAN
than with KDMD in the most turbulent setting. We conjecture that this is due to
the LRAN's flexibility in learning complicated observables from data, thereby
serving as a viable surrogate model for the main structure of fluid dynamics in
turbulent convection settings. In contrast, KDMD was more effective in lower
turbulence settings due to the repetitiveness of the convection flow. The
feasibility of Koopman-based surrogate models for turbulent fluid flows opens
possibilities for efficient model-based control techniques useful in a variety
of industrial settings.
