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.

通过 Koopman-Based 机器学习结构和线性循环自编码网络（LRAN），我们研究了 Rayleigh-Bénard 对流系统中湍流对流流体流动的简化动力学模型以及其应用于模型预测与控制技术。

基于 Koopman 的湍流 Rayleigh-Bénard 对流的代理模型

Koopman-Based Surrogate Modelling of Turbulent Rayleigh-Bénard  Convection

Complex spatial and temporal structures are inherent characteristics of
turbulent fluid flows and comprehending them poses a major challenge. This
comprehesion necessitates an understanding of the space of turbulent fluid flow
configurations. We employ a diffusion-based generative model to learn the
distribution of turbulent vorticity profiles and generate snapshots of
turbulent solutions to the incompressible Navier-Stokes equations. We consider
the inverse cascade in two spatial dimensions and generate diverse turbulent
solutions that differ from those in the training dataset. We analyze the
statistical scaling properties of the new turbulent profiles, calculate their
structure functions, energy power spectrum, velocity probability distribution
function and moments of local energy dissipation. All the learnt scaling
exponents are consistent with the expected Kolmogorov scaling and have lower
errors than the training ones. This agreement with established turbulence
characteristics provides strong evidence of the model's capability to capture
essential features of real-world turbulence.

利用基于扩散的生成模型学习湍流涡度轮廓的分布，生成与训练数据集不同的多样化湍流解，并分析新湍流轮廓的统计缩放特性、能量功率谱、速度概率分布函数和局部能量耗散矩。通过与已建立的湍流特性的一致性，该模型证明了其捕捉实际湍流关键特征的能力。