## 1 Introduction  
Efficient representation of high-dimensional spatiotemporal flow fields and accurate capture of their dynamic evolution have long posed significant challenges in computational fluid dynamics (CFD). This is especially true for flow systems, such as cylinder wake flows [1, 2], which exhibit both classical and complex behaviors. Although traditional CFD methods deliver high computational accuracy, their substantial computational costs and storage requirements severely limit their applicability in real-time prediction and optimization. To overcome these limitations, a wide range of model reduction strategies--broadly referred to as reduced-order models (ROMs) [3, 4]--have been used to construct low-dimensional representations that capture dominant flow features. The fundamental strength of such approaches lies in establishing a robust mapping between high-dimensional dynamical systems and their low-dimensional surrogates--a task that in volves not only compressing data but also uncovering the essential physical mechanisms underlying fluid motion. Traditional ROM methods can be broadly categorized into three types:  
1. Modal decomposition projection methods (e.g., POD-Galerkin [5]) construct low-dimensional subspaces by extracting dominant modes and projecting governing equations onto these subspaces. 2. Balanced truncation methods [6] utilize controllability and observability analysis to retain state variables most significant to input-output behavior. 3. Harmonic balance methods [7] address periodic flows by approximating steady-state solutions in the frequency domain using a finite set of Fourier basis functions. These approaches are physically grounded and interpretable, offering energy-optimal modal selection and high-fidelity reconstruction. However, they often rely on linear subspace assumptions and require precise numerical integration, which can be limiting for nonlinear or geometrically complex flows. Beyond traditional projection-based approaches, data-driven techniques have enabled a broader exploration of flow structures without reliance on governing equations. These can be classified into:  
1. Data-driven modal methods, like snapshot POD [8] and dynamic mode decomposition (DMD) [9], which extract coherent structures directly from simulation or experimental data. They are equation-free but often sensitive to noise and require extensive datasets. 2. Neural-network-based approaches, such as autoencoders [10] and their derivatives [11], which learn nonlinear latent representations but may suffer from overfitting and lack interpretability. 3. Hybrid strategies combining ROM and machine learning, such as POD-RBF, POD-LSTM [12], and physics-informed neural networks (PINNs) [13; 14], aiming to balance physical relevance and learning flexibility. In addition to standard POD, various studies have expanded its framework:  
* [PERSON] [15] proposed a robust spectral POD (SPOD) integrating robust principal component analysis for improved noise resilience. * [PERSON] [16] introduced frequency-domain POD (FD-POD) to reconstruct unsteady flows using embedded frequency information. * [PERSON] [17] developed POD-ISAT, combining POD with in situ adaptive tabulation for efficient steady-state PDE approximation. Snapshot POD remains a popular tool due to its algorithmic simplicity and modal resolution, but its linear nature limits its ability to capture nonlinear interactions, such as turbulence. This motivates the search for more expressive and adaptive dimensionality reduction techniques. Among these, clustering-based methods have emerged as powerful alternatives for representing nonlinear, multi-regime flow behaviors. Instead of relying on orthogonal basis functions, they group data points by similarity in feature space, making them well-suited to reflect local state transitions and complex dynamics. This is particularly useful in transitional flows, where traditional modal decompositions fall short.