## 2 C-POD Method and Modal Sorting Method  
### C-POD Method  
The proper orthogonal decomposition (POD) method, based on singular value decomposition (SVD), is a well-established, data-driven model reduction technique. POD generates a set of orthogonal basis functions that encapsulate the dominant features of a system. By employing linear combinations of these selected basis functions, one can effectively approximate the original system, thereby transforming a high-dimensional problem into a low-dimensional representation. To preserve computational resources, the snapshot POD method proposed by [PERSON] is frequently used for dimensionality reduction of the dataset [23]. However, POD is inherently a linear reduction technique. Although it captures the primary dynamic features of linear systems, it may not fully represent the complexities of nonlinear flows, particularly when nonlinear behaviors vary across multiple time steps and spatial domains. In such cases, the low-rank approximation afforded by POD might not sufficiently capture these nonlinear effects. Figure 1: Taxonomy of ROM. Clustering methods exhibit robust capabilities in extracting nonlinear features, effectively capturing patterns and regularities in data, especially within complex nonlinear systems. Moreover, these methods can autonomously adapt to diverse data patterns, mapping both linear and nonlinear datasets into a low-dimensional space that elucidates underlying dynamic characteristics. Consequently, this paper integrates clustering concepts with model reduction techniques by proposing a clustering-based dimensionality reduction framework, referred to as C-POD, to enhance the effectiveness of proper orthogonal decomposition (POD) in addressing nonlinear model reduction challenges. The construction process is detailed as follows:  
1. Arrange the responses of \(m\) data points under \(n\) different operating conditions as columns to form the database matrix \(\mathbf{D}\): \[\mathbf{D}=\{d_{1},d_{2},\ldots,d_{n}\}\in R^{m\times n}\] (1) Here, \(d_{i}\in R^{m}\) (\(i\) = 1, 2, \(\ldots\), \(n\)) represents the data snapshot under the \(i\)-th operating condition. 2. Determining initial cluster centers based on the snapshot POD method [23]. First, the correlation matrix of the snapshot matrix \(\mathbf{D}\) is constructed: \[\mathbf{C}=\mathbf{D}\mathbf{D}^{T}\in R^{m\times m}\] (2) Solving the eigenvalue problem of the correlation matrix \(\mathbf{C}\): \[\mathbf{\varphi}_{i}=\lambda_{i}\mathbf{\varphi}_{i}\ (i=1,2,\ldots m)\] (3) Here, \(\lambda_{i}\) represents the eigenvalue, and \(\mathbf{\varphi}_{i}\) is the corresponding eigenvector. The eigenvalues are arranged in descending order: \[\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{m}\geq 0\] (4) The corresponding eigenvectors \(\mathbf{\varphi}_{i}\) form an orthogonal basis, and the first \(K\) eigenvectors are selected as the initial cluster centers: \[\mathbf{\mu}_{i}{}^{(0)}=\mathbf{\varphi}_{i}(i=1,2,\ldots K)\] (5) 3. Sample assignment. Each column vector \(d_{j}\) (\(j\) = 1, 2, \(\ldots\), \(n\)) from the data matrix \(\mathbf{D}\) is assigned to the nearest cluster center, forming \(K\) clusters: \[\mathbf{S}_{i}^{(t)}=\{\mathbf{d}\in\mathbf{D}\Big{\|}\Big{\|}\mathbf{d}-\mathbf{\mu}_{i}^{(t-1)} \Big{\|}^{2}\leq\Big{\|}\mathbf{d}-\mathbf{\mu}_{j}^{(t-1)}\Big{\|}^{2},\forall j\
eq i\}\] (6) Here, \(\mathbf{S}_{i}^{(t)}\) represents the number of vectors in the \(i\) cluster, \(\mathbf{\mu}_{i}\) denotes the \(i\)-th cluster centroid, \(t\) is the current iteration count, and \(|\ |\cdot|\ |\) represents the Euclidean distance norm. 4. Updating the cluster centers. The cluster centers are updated by calculating the mean of all vectors in each cluster \(\mathbf{S}_{i}\), and this mean becomes the new cluster centroid: \[\mathbf{\mu}_{i}^{(t)}=(\frac{1}{\big{|}\mathbf{S}_{i}^{(t)}\big{|}}){\sum}_{x\in \mathbf{S}_{i}^{(t)}}\,x(i=1,2,\ldots,K)\] (7) where \(\mathbf{S}_{i}^{(t)}\) denotes the set of snapshots assigned to cluster \(i\).