Koopman-Based Surrogate Modelling of Turbulent Rayleigh-Bénard Convection

TL;DR

This work compares Koopman-based surrogate models for turbulent Rayleigh-Bénard Convection using a Linear Recurrent Autoencoder Network (LRAN) against Kernel Dynamic Mode Decomposition (KDMD). Across four Rayleigh numbers spanning increasing turbulence, LRAN achieves higher predictive accuracy in the most turbulent regimes, while KDMD excels at the least turbulent case due to its capacity to capture repetitive dynamics. The study systematically tunes hyperparameters for both methods and demonstrates that LRAN's flexible observable space can better represent non-periodic, turbulent flow features, with potential applicability to model-based control of convection. The results suggest Koopman-based surrogates are a viable path toward efficient, data-driven control of complex fluid flows, motivating further physics-informed enhancements and controlled RBC extensions.

Abstract

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 Bénard 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.

Figures (6)

• Figure 1: Left: Rayleigh Benard Convection state. Color is temperature, arrows fluid velocity. Right: Corresponding local convective field. The state results from a Direct Numerical Simulation for Rayleigh number $Ra=2e6$.
• Figure 2: Illustration of the Linear Recurrent Autoencoder Network architecture. Starting from the bottom left the encoder $f_{enc, \theta}$ compresses snapshot $q_t$ to observables $g$. The matrix K evolves the observable forward in time. The decoder $f_{dec, \theta}$ lifts the observables back into the original space to obtain predictions $\hat{q}$. Grayed-out snapshot compressions $q_{t+1}$ and subsequent are only used during training.
• Figure 3: Correlation of hyperparameters to the NSSE on the test data. Panel (a) shows the correlation of LRAN sequence length and test NSSE. Each point represents a training run in the LRAN sweep for $Ra=1e6$. Panel (b) shows the NSSE for tested KDMD snapshot size values grouped by $Ra$.
• Figure 4: The evolution of the NSSE in the test sequence averaged over all episodes. Shaded regions indicate the standard deviation.
• Figure 5: Examples for ground truth and LRAN predicted local convective fields for $Ra=5e6$.