## 4 Training of MIMO-NARX Meta-Models
### MIMO-NARX Model Training  
Firstly, the MOR strategy is applied to transform the nine-DOF to the reduced coordinates for the convenience of MIMO-NARX modeling. The snapshot matrix \(\mathbf{X}\) is extracted from the structural responses of the two hundred dynamic history simulations. The transformation matrix \(\mathbf{\Phi}_{n_{r}}\) is obtained through SVD as the first \(n_{r}\) POD modes according to Equations (9) and (10). The coordinate is transformed from the original order \(n=9\) to the reduced order \(n_{r}=4\), which is determined by the energy criteria in Equation (11). The fitness indicator of the nine-DOF response in original coordinates and reverse-transformation response is shown in Figure 6(a), in which the red line indicates the median, the blue box indicates the 25 th and 75 th percentiles, and the red plus symbol indicates outliers. It can be observed from Figure 6(a) that the maximum error of MOR occurs in the first DOF because the value of absolute response of the first floor is the smallest compared with the other floors. For a randomly selected sample, the comparison in Figure 6(b) shows a good fitness of 92% between original and reverse-transformation velocity responses \(\dot{\mathbf{x}}(t)\) and \(\dot{\mathbf{x}}_{-}(t)\) for the first floor. This implies that the error introduced by MOR is negligible in this study. To capture the nonlinear dynamic behavior of the MDOF system, the MIMO-NARX model is trained from the \(n_{s}=200\) dynamic history simulation samples. Multiple inputs and outputs in the reduced coordinates are then used to build the MIMO-NARX model to avoid significant computational efforts associated with the original number of DOFs. The simulation samples are first examined by a predefined threshold value to screen responses of interest that exhibit suitable nonlinearity [18]. NARX candidates are identified by the LARs algorithm and the corresponding term coefficients are calculated by the OLS approach as shown in Figure 8. The optimal NARX model is selected based on the minimum mean errors of these candidates and terms pruning is then applied to improve precision. If the accuracy requirement is not satisfied for the optimal NARX model, some measures to improve NARX accuracy need to be taken such as redesigning the formulation of the NARX full model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & **Distribution** & \multirow{2}{*}{**Bounds**} & \multirow{2}{*}{**Mean**} & **Standard** \\  & **Type** & & & **Deviation** \\ \hline \(\omega_{mid}/2\pi\) & Gamma & [0, +\(\infty\)] & 5.87 & 3.11 \\ \(\zeta_{f}\) & Beta & [0.02, 1] & 0.213 & 0.143 \\ Mass 1 & Gaussian & [392.77, 729.43] & 561.1 & 56.11 \\ Mass 2 & Gaussian & [384.58, 714.22] & 549.4 & 54.94 \\ Mass 3 & Gaussian & [416.08, 772.72] & 594.4 & 59.44 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Marginal distributions of uncertainties. Figure 7: MOR error between the original and reverse-transformation responses. (**a**) Fitness for nine DOFs (200 samples); (**b**) 1 st DOF response (a random sample). In this study, the MIMO-NARX full model is designed as shown in Figure 9. The ground motion accelerations are considered as the input, while responses of the \(n_{r}=4\) MOR reduced coordinates are used as the outputs. The maximum time lag of \(n_{i}=n_{o}=4\) is used for the input and output in the NARX model to capture the memories. The current value for each output is replicated by one MIMO-NARX model considering the coupling of the past signals of all four outputs. The velocity responses in the reduced coordinates are selected as NARX outputs because previous studies showed that it is more effective to describe nonlinear systems with respect to velocity.