## 3 Model-Oder Reduction
### Reduced Model Based DMDc  
To identify the dynamics in terms of the low-dimensional state representation \(\mathbf{\xi}\in\mathbb{R}^{q}\) (\(q\ll N\)) by means of DMDc [13], consider the snapshot and input matrices  
\[\Xi= \begin{bmatrix}\mathbf{\xi}(0)&\mathbf{\xi}(1)&\cdots&\mathbf{\xi}(m) \end{bmatrix} \in\mathbb{R}^{q\times m},\quad\Xi^{+}= \begin{bmatrix}\mathbf{\xi}(1)& \cdots&\mathbf{\xi}(m+1)\end{bmatrix} \in\mathbb{R}^{p\times m},\] \[U= \begin{bmatrix}u(0)&\cdots&u(m)\end{bmatrix} \in\mathbb{R}^{1 \times m},\quad\Xi_{u}= \begin{bmatrix}\Xi\\ U\end{bmatrix} \in\mathbb{R}^{(q+1)\times m}\]  
Figure 3: (**Left**): Comparison of the original data set (top) and the approximated one obtained with the trained autoencoder (bottom). (**Right**): Approximation error norm. Figure 2: Autoencoder network for empirical basis identification using machine learning. with their respective singular value decompositions  
\[\Xi_{u} =U_{1}\Sigma_{1}V_{1}^{\mathrm{T}},\quad\Xi^{+}=U_{2}\Sigma_{2}V_{2} ^{\mathrm{T}},\quad U_{k}=\big{[}U_{k,r}\quad U_{k,0}\big{]},\] \[\Sigma_{k} = \begin{bmatrix}\Sigma_{k,r}&0\\ 0&\Sigma_{k,0}\end{bmatrix} ,\quad V_{k}= \begin{bmatrix}V_{k,r}\\ V_{k,0}\end{bmatrix} ,\quad k=1,2,\]  
where  
\[U_{1} \in\mathbb{R}^{(q+1)\times(q+1)},\quad U_{1,r}\in\mathbb{R}^{(q+ 1)\times r_{1}},\quad U_{1,0}\in\mathbb{R}^{(q+1)\times(q+1-r_{1})}\] \[\Sigma_{1} \in\mathbb{R}^{(q+1)\times(q+1)},\quad\Sigma_{1,r}\in\mathbb{R}^ {r_{1}\times r_{1}},\quad\Sigma_{1,0}\in\mathbb{R}^{(q+1-r_{1})\times(q+1-r_{ 1})}\] \[V_{1} \in\mathbb{R}^{m\times(q+1)},\quad V_{1,r}\in\mathbb{R}^{m\times r _{1}},\quad V_{1,0}\in\mathbb{R}^{m\times(q+1-r_{1})}\] \[U_{2} \in\mathbb{R}^{q\times q},\quad U_{2,r}\in\mathbb{R}^{q\times r _{2}},\quad U_{2,0}\in\mathbb{R}^{q\times(q-r_{2})}\] \[\Sigma_{2} \in\mathbb{R}^{q\times q},\quad\Sigma_{2,r}\in\mathbb{R}^{r_{2} \times r_{2}},\quad\Sigma_{2,0}\in\mathbb{R}^{(q-r_{2})\times(q-r_{2})}\] \[V_{2} \in\mathbb{R}^{m\times q},\quad V_{2,r}\in\mathbb{R}^{m\times r _{2}},\quad V_{2,0}\in\mathbb{R}^{m\times(q-r_{2})}.\]  
Considering that  
\[\xi(t+1)=A\xi(t)+bu(t)= \begin{bmatrix}A\quad\mathbf{b}\end{bmatrix} \begin{bmatrix}\xi(t)\\ u(t)\end{bmatrix} \]  
one obtains  
\[\Xi^{+}= \begin{bmatrix}A\quad\mathbf{b}\end{bmatrix} \Xi_{u}\]  
or equivalently  
Taking into account only the dominant singular values \(\sigma_{2,i},i=1,\ldots,r_{2}\) and \(\sigma_{1,j},\ j=1,\ldots,r_{1}\), this becomes  
\[U_{2,r}\Sigma_{2,r}V_{2,r}^{\mathrm{T}}\approx \begin{bmatrix}A\quad\mathbf{b} \end{bmatrix} U_{1,r}\Sigma_{1,r}V_{1,r}^{\mathrm{T}},\]  
yielding  
\[ \begin{bmatrix}A\quad\mathbf{b}\end{bmatrix} \approx U_{2,r}\Sigma_{2,r}V_{2,r}^{ \mathrm{T}}V_{1,r}\Sigma_{1,r}^{-1}U_{1,r}^{\mathrm{T}}.\]  
Introducing the reduced order state  
\[\mathbf{z}(t)=U_{2,r}^{\mathrm{T}}\xi(t)\in\mathbb{R}^{r_{2}},\quad\xi(t)=U_{2,r} \mathbf{z}(t)\]  
and noticing that \(U_{2,r}^{\mathrm{T}}U_{2,r}=I_{q}\) it holds that  
\[\mathbf{z}(t+1)=U_{2,r}^{\mathrm{T}} \begin{bmatrix}A\quad\mathbf{b}\end{bmatrix} \begin{bmatrix}\xi(t)\\ u(t)\end{bmatrix} \approx\Sigma_{2,r}V_{2,r}^{\mathrm{T}}V_{1,r}\Sigma_{1,r}^{-1 }U_{1,r}^{\mathrm{T}} \begin{bmatrix}\xi(t)\\ u(t)\end{bmatrix} .\]  
Thus the low-dimensional dynamic model is obtained in the form  
\[\mathbf{z}(t+1)=\bar{A}\mathbf{z}(t)+\bar{\mathbf{b}}u(t)\]with  
\[\tilde{A} =U_{2,r}^{\rm T}AU_{2,r}\approx\left[\Sigma_{2,r}V_{2,r}^{\rm T}V_{1,r }\Sigma_{1,r}^{-1}U_{1,r}^{\rm T}\right]_{(;1;1q)}U_{2,r},\] \[\tilde{\mathbf{b}} =U_{2,r}^{\rm T}\mathbf{b}\approx\left[\Sigma_{2,r}V_{2,r}^{\rm T}V_{1,r}\Sigma_{1,r}^{-1}U_{1,r}^{\rm T}\right]_{(;q+1)}\]  
The resulting network-based model order reduction scheme is shown in Figure 4.  
The obtained results with \(r_{2}=17\), \(r_{1}=18\) are shown in Figure 5. On the left of the figure, the comparison of the reference solution and the autoencoder-DMDc (ae-dmdc) based approximation can be seen. On the right part of the figure, the associated approximation error norm is shown.