## 2 Methodology Details
### DMD Method  
The DMD method put forward by [PERSON] [38] can extract valuable information from the flow field snapshots obtained from computation or experimentation. The snapshot sequence is represented by the matrix \(V_{0}^{N}\). \[V_{0}^{N}=\{v_{0},v_{1},v_{2},v_{3},\ldots,v_{N}\} \tag{13}\]  
The snapshot sequence is separated by a sampling time interval \(\Delta t\), and \(v_{i}\) indicates the i-th data, while subscript 0 and \(N\) represent the first and last element in the sequence. If the sampling interval is small enough, the data sequence can be regarded as linear dynamical systems. \[v_{i+1}=Av_{i} \tag{14}\]  
Here, \(A\) is a constant matrix. Moreover, the following equation can be deduced. \[V_{1}^{N}=AV_{0}^{N-1} \tag{15}\]  
Next, an eigenvalue decomposition of \(A\) is performed with \(\Lambda\) as its eigenvalue and \(S\) as its eigenvector. \[A=S^{-1}\Lambda S \tag{16}\]  
Since the dimension of the columns of matrix A are much larger than that of the rows, it is very complicated to directly perform singular value decomposition (SVD), so the similarity transformation of \(A\) is applied to obtain the similarity matrix of \(A\), which simplifies the solution of eigenvalues and eigenvectors. \[V_{0}^{N-1}=U\sum D^{*} \tag{17}\]  
\[A=V_{1}^{N}D\sum^{-1}U^{*} \tag{18}\]  
\[\widetilde{A}=U^{*}AU=U^{*}V_{1}^{N}D\sum^{-1} \tag{19}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Allowable** & **Grid** & **Number of** & **Accunulated** & **Allowable Number** & **Reliability** \\
**Error (\%)** & **Resolution** & **Time Steps** & **Error** & **of Time Steps** & **Reliability** \\ \hline
5 & 3000 \(\times\) 200 & 72,000 & 9.97 \(\times\) 10\({}^{-6}\) & 1.81 \(\times\) 10\({}^{12}\) & 2.51 \(\times\) 10\({}^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Error estimates. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**N1, N2, N3** & \(r_{21}\) & \(r_{32}\) & \(\phi_{1}\) & \(\phi_{2}\) & \(\phi_{3}\) & P & \(e_{a}^{32}\) & \(GCI_{fine}^{32}\) \\ \hline
750,000, 600,000, 300,000 & 1.118 & 1.414 & 984.422 & 984.3101 & 983.8613 & 1.09 & 0.046\% & 0.12\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Sample calculation of discretization error. The eigenvalues of \(A\) are the same as those of \(\widetilde{A}\), and the eigenvectors of \(A\) have a transformation relationship with those of \(\widetilde{A}\). Therefore, the eigenvalues and eigenvectors of \(A\) can be obtained by performing SVD on \(\widetilde{A}\). After the above solution process, \(S^{-1}\) and \(\Lambda\) are expressed as follows. \[S^{-1}=[y_{1},y_{2},\dots,y_{n}] \tag{20}\]  
\[\Lambda= \begin{bmatrix}\lambda_{1}&0&\cdots&0\\ 0&\lambda_{2}&\ddots&\vdots\\ \vdots&\ddots&&\ddots&0\\ 0&0&&\dots&\lambda_{r}\end{bmatrix}  \tag{21}\]  
Matrix \(W_{i}\) is introduced here. \[W_{i}=Sv_{i} \tag{22}\]  
The following expression can be derived from the above formula. \[v_{i}=S^{-1}W_{i}=S^{-1}\wedge^{i}W_{0} \tag{23}\]  
\[W_{0}={(S^{-1})}^{-1}v_{0} \tag{24}\]  
Thus, the flow field at any time can be obtained by:  
\[v_{i}=\sum_{k=1}^{r}y_{k}\lambda_{k}^{i}W_{k}^{0} \tag{25}\]  
Here, \(W_{k}^{0}\) is the k-th element of vector \(W_{0}\). For an approximately linear system, there is a certain error in using Equation (24) to calculate \(W_{0}\), so the least squares method is used to calculate \(W_{0}\). \[\underset{F}{minimize}\quad J(W_{0}):=\|V_{1}^{N}-U\widetilde{A}\Sigma D^{*} \|_{F}^{2} \tag{26}\]  
Next, the data sequence can be described by the following expression. \[[v_{0},v_{1},v_{2},\dots,v_{n}]=[y_{1},y_{2},\dots,y_{r}] \begin{bmatrix}W_{1} ^{0}&\\ &W_{2}^{0}&\\ &\ddots&\\ &&W_{r}^{0}\end{bmatrix}\begin{bmatrix}1&\lambda_{1}&\cdots&\lambda_{1}^{n}\\ 1&\lambda_{2}&\ddots&\vdots\\ \vdots&\cdots&\\ 1&\lambda_{r}&\cdots&\lambda_{r}^{n}\end{bmatrix}  \tag{27}\]  
In this way, the DMD method completes the decomposition of the flow field.