## 2 Preliminary Theory and Related Work
### Singular Value Decomposition (SVD)  
Singular Value Decomposition (SVD) is a well-established and extensively studied method. SVD decomposes the snapshot matrix to obtain the optimal orthogonal basis and projects each column vector onto this orthogonal basis to obtain the corresponding eigenmodes. Specifically, for any matrix \(m\times n\)\(\mathbf{H}\), SVD decomposes it into the product of three special matrices:  
\[\mathbf{H}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{*}, \tag{15}\]  
where:  
\(\mathbf{U}\) is an \(m\times m\) orthogonal matrix. The columns \(\mathbf{u}_{i},i=1,2,\dots,k\), are called left singular vectors. They form an orthonormal basis for the column space of \(\mathbf{H}\) and can also be viewed as the eigenfunctions of the Koopman operator in the \(m\)-step delay coordinates. \(\mathbf{V}\) is an \(n\times n\) orthogonal matrix. The columns \(\mathbf{v}_{i}\) are called right singular vectors and are the principal modes in the space of the observable functions \(\mathbf{u}_{i}\) (basis functions) corresponding to the Koopman modes. \(\mathbf{\Sigma}\) is an \(m\times n\) diagonal matrix whose non-negative real entries are called singular values. These values, often referred to as \"energy\", are usually arranged in descending order. By applying SVD, the \(i\)-th column of the matrix \(\mathbf{H}\) can be represented as:  
\[\mathbf{H}_{i}=\mathcal{K}^{i-1}f=\sum_{j=1}^{n}\sigma_{j}\mathbf{u}_{j} \mathbf{v}_{ji}.\]  
A key feature of SVD is that it orders the orthogonal bases and their corresponding eigenmodes in descending order according to their \"energy\" values. **Theorem 1**. : _Suppose \(\mathbf{H}_{n}\in\mathbb{R}^{n\times n}\) is a positive Hankel matrix with singular values \(\sigma_{1},\sigma_{2},\dots,\sigma_{n}\). The theorem indicates:_  
\[\sigma_{j}\leq C\rho^{-j}log^{-1}\sigma_{1}\]  
_for constants \(C\) and \(\rho\), and for \(j=1,\dots,n\)._This theorem demonstrates that a Hankel matrix can be well-approximated by a low-rank matrix, meaning that the columns in the state snapshot matrix can be represented as linear combinations of a few eigenvectors [21]. The total energy of the first \(k\) modes is defined as:  
\[E(k)=\frac{\sum_{i=1}^{k}\sigma_{i}}{\sum_{i=1}^{N}\sigma_{i}} \tag{16}\]  
This indicates the contribution of the first \(k\) eigen time series to the total energy. The first few modes typically capture the majority of the information in the flow field and are sufficient for reconstructing the flow field with adequate accuracy. Therefore, mode truncation is commonly performed where the modal energy reaches 99% of the total energy. Regarding the choice of \(r\): When the singular values decay rapidly, a smaller \(r\) is typically chosen because the majority of the energy is concentrated in the first few modes. The most common approach to selecting \(r\) is to choose the value of \(r\) that allows \(E(k)\) to reach the threshold of 99%. In some cases, the optimal \(r\) can also be determined by calculating the reconstruction error for different values of \(r\). In practical applications, the selection of \(r\) also depends on prior knowledge of the system and experimental experience. In this paper, we will determine the truncation order \(r\) by considering the variation of \(E(k)\) with \(k\), reconstruction accuracy, and reconstruction performance. Once \(r\) is selected, we will refer to the modes as forced eigenmodes (the term \"forced\" will be explained later). Furthermore, the first two columns of the \(\mathbf{V}\) matrix obtained by the SVD algorithm correspond to the variations of each column of the \(\mathbf{H}\) matrix projected onto the two orthogonal bases with the highest energy. These two dominant modes \((v_{1},v_{2})\) capture the main dynamical behavior of the system in a two-dimensional embedded space. Through a linear combination of these two modes, the system's trajectory in the two-dimensional embedded space can be approximately reconstructed.