## 2 Materials and Methods
### Proposed Methodology
#### 2.3.1 Data Preparation  
The extracted data from the finite elements model are the initial nodes coordinates and their displacements after each operation. In our case, we want to use the surrogate model to generate all the intermediate structure states for a new sequence \(S_{k}\) (\(k\) is the sequence number in the DoE) of size \(T\) and initial state \(X_{k}(t_{0})=X_{0}\). \(T\) is the number of unit operations, \(T=8\) for all sequences. Thus, for all \(i\in\{1,\ldots,T\}\), we need to predict the state \(X_{k}(t_{i})\) taking as input the previous state \(X_{k}(t_{i-1})\) when applying the unitary operation \(s_{k}(t_{i-1})\). Thus, the regression reads as:  
\[X_{k}(t_{i})=f(X_{k}(t_{i-1}),s_{k}(t_{i-1})),\ \forall i\in\{1,\ldots,T\}. \tag{10}\]  
Because the function we search must be the same for all states of the system, we can write Equation (10) by grouping all states of the system in the same matrix, as it is done when applying the DMD  
\[\mathbb{X}_{k}=f\big{(}\mathbb{\hat{X}}_{k},\ \mathbb{\hat{s}}_{k}\big{)}, \tag{11}\]  
with \(\mathbb{\hat{X}}_{k}=[X_{k}(t_{T-1})\ \cdots\ X_{0}]\) and \(\mathbb{X}_{k}=[X_{k}(t_{T})\ \cdots\ X_{k}(t_{1})]\) respectively the input and the output state matrices (of size \(N\times T\)) of the system subjected to the sequence \(S_{k}\). Figure 4: Final state related to an 8-unitary operations sequence (displacements \(\times\)3). Once again, because the function \(f\) must be the same for all tested sequences, i.e., the whole DoE, constituted of \(N_{\text{C}}\) simulations, we can concatenate all the input matrices together and all the output matrices together into bigger matrices that we simply note \(\mathbb{X}\) and \(\mathbb{X}\), both of size \(N\times(T\times N_{\text{C}})\). Thus, the regression problem reads  
\[\mathbb{X}=f\left(\mathbb{X},\,S\right), \tag{12}\]  
where \(S\) is the vector concatenating all the sequences, of size \(T\times N_{\text{C}}\) and \(f\) the generic function. This function is constructed (learned from the data) by using the procedures described in Section 2.3.2. Figure 5: High-fidelity simulation of the sequence \(\{1,2,1,2,1,1,2,2\}\) (displacements \(\times 3\) for visualization). Both stiffeners are placed from the beginning on the sheet, and each layer of them is sequentially mechanically activated. In order to reduce the complexity, we compute the SVD of matrix \(\mathbb{X}\), built for each coordinate axis. The normalized singular values related to the three nodal displacements are plotted in Figure 6.  
As we can see in these graphs, the singular values decrease very fast, and three modes suffice to reduce the singular values of two orders of magnitude. Thus, we expect approximating reasonably well the structure state (nodal displacements) with a very limited number of vectors of the reduced basis. In addition, 95% of accuracy is attained by employing only the first three modes that will constitute the reduced basis. This reduction is quite impressive: moving from a space of dimension \(N\) (the number of nodes) to one involving only three degrees of freedom (the size of the reduced basis) only introduced a 5% loss of accuracy. In what follows, we consider richer bases for obtaining higher accuracy. Figure 7 plots the approximation error of all the states concerned in the DoE projected into a 5D reduced basis. Even if, as noticed, 5 modes offer excellent performances, errors in the prediction are amplified in posterior operations, and, consequently, it is important to retain a reduced basis that is accurate enough. Figure 6: Normalized singular values related to the three nodal displacements. Figure 7: Graph: Maximum modal approximation error with respect to the maximum displacement when using five modes. Picture: Reduced approximation of the deformed structure (displacements \(\times 3\)). * _MLP-based regression._  
To set the MLP hyper-parameters, we iterate until obtaining results good enough in the approximation function.