Numerically solving a large parametric nonlinear dynamical system is challenging due to its high complexity and the high computational costs. In recent years, machine-learning-aided surrogates are being actively researched. However, many methods fail in accurately generalizing in the entire time interval $[0, T]$, when the training data is available only in a training time interval $[0, T_0]$, with $T_0

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.

Koopman decomposition is a non-linear generalization of eigen decomposition, and is being increasingly utilized in the analysis of spatio-temporal dynamics. Well-known techniques such as the dynamic mode decomposition (DMD) and its variants provide approximations to the Koopman operator, and have been applied extensively in many fluid dynamic problems. Despite being endowed with a richer dictionary of nonlinear observables, nonlinear variants of the DMD, such as extended/kernel dynamic mode decomposition (EDMD/KDMD) are seldom applied to large-scale problems primarily due to the difficulty of discerning the Koopman invariant subspace from thousands of resulting Koopman triplets: eigenvalues, eigenvectors, and modes. To address this issue, we revisit the formulation of EDMD and KDMD, and propose an algorithm based on multi-task feature learning to extract the most informative Koopman invariant subspace by removing redundant and spurious Koopman triplets. These algorithms can be viewed as sparsity promoting extensions of EDMD/KDMD and are presented in an open-source package. Further, we extend KDMD to a continuous-time setting and show a relationship between the present algorithm, sparsity-promoting DMD and an empirical criterion from the viewpoint of non-convex optimization. The effectiveness of our algorithm is demonstrated on examples ranging from simple dynamical systems to two-dimensional cylinder wake flows at different Reynolds numbers and a three-dimensional turbulent ship air-wake flow. The latter two problems are designed such that very strong transients are present in the flow evolution, thus requiring accurate representation of decaying modes.