Reduced order models (ROMs) that capture flow dynamics are of interest for decreasing computational costs for simulation as well as for model-based control approaches. This work presents a data-driven framework for minimal-dimensional models that effectively capture the dynamics and properties of the flow. We apply this to Kolmogorov flow in a regime consisting of chaotic and intermittent behavior, which is common in many flows processes and is challenging to model. The trajectory of the flow travels near relative periodic orbits (RPOs), interspersed with sporadic bursting events corresponding to excursions between the regions containing the RPOs. The first step in development of the models is use of an undercomplete autoencoder to map from the full state data down to a latent space of dramatically lower dimension. Then models of the discrete-time evolution of the dynamics in the latent space are developed. By analyzing the model performance as a function of latent space dimension we can estimate the minimum number of dimensions required to capture the system dynamics. To further reduce the dimension of the dynamical model, we factor out a phase variable in the direction of translational invariance for the flow, leading to separate evolution equations for the pattern and phase. At a model dimension of five for the pattern dynamics, as opposed to the full state dimension of 1024 (i.e. a 32x32 grid), accurate predictions are found for individual trajectories out to about two Lyapunov times, as well as for long-time statistics. Further small improvements in the results occur at a dimension of nine. The nearly heteroclinic connections between the different RPOs, including the quiescent and bursting time scales, are well captured. We also capture key features of the phase dynamics. Finally, we use the low-dimensional representation to predict future bursting events, finding good success.

翻译：降階模型（ROMs）能夠捕捉流體動力學特性，對於降低模擬計算成本及實現基於模型的控制方法具有重要意義。本研究提出一個數據驅動框架，用以構建有效捕捉流體動力學特性與性質的最小維度模型。我們將其應用於科爾莫戈羅夫流中一個包含混沌與間歇行為的流態——此類行為常見於多種流體過程且難以建模。流體的運動軌跡接近相對週期軌道（RPOs），並夾雜著對應於不同RPO區域間遊走所導致的零星爆發事件。模型構建的第一步是利用欠完備自編碼器將全狀態數據映射至維度顯著降低的潛在空間；隨後建立潛在空間中離散時間動力學的演化模型。通過分析模型性能隨潛在空間維度的變化，可估算捕捉系統動力學所需的最小維度。為進一步降低動力學模型維度，我們分離出流體平移不變方向上的相位變量，導出模式與相位的獨立演化方程。當模式動力學維度降至五維（而全狀態維度為1024，即32×32網格）時，單個軌跡在約兩個Lyapunov時間內的預測以及長時間統計特性均能準確呈現；當維度增至九維時結果僅有微小改善。模型成功捕捉了不同RPO間近異宿連接（包括靜默期與爆發期的時間尺度），亦保留相動力學的關鍵特徵。最終，我們利用低維表徵預測未來爆發事件，取得了良好成效。