## 3 Application Example
### Construction of ANN-Based Surrogate Model  
In this section, an ANN-based surrogate model was constructed to predict the TSTT performance of the bridge transport network. Five damage states of 48 bridges were considered as the input values of the ANN model for constructing the surrogate model and were assumed to be integer values from 0 to 4 depending on each damage state. In addition, for the input data generation for training, Epicenter 8 shown in Figure 5 was selected as the training epicenter, and the input earthquake magnitude was  
Figure 5: Locations of the considered bridge structures and epicenters in the target network. Figure 6: Discrete bounded probability density function of earthquake magnitude ranging from 4.5 to 7.5.  
considered as 7.0 with a focal depth of 10 km. Moreover, the ground motion intensity SA of each bridge was utilized to determine the damage state of each bridge according to the spatially correlated seismic attenuation law from the input earthquake. Finally, TSTT values for each scenario were calculated by EMFE/4 and considered as output data. In addition, to reflect the traffic flow capacity of the damaged bridge, the reduced traffic volume was considered according to each damage state. Table 1 shows the reduced capacities of a damaged bridge structure according to damage states for transportation network analysis. In this study, an ANN-based surrogate model was trained with a total of 100,000 randomly sampled data. In addition, 30,000 sufficient epochs were adopted, and 100,000 data were set with the ratio 0.7:0.15:0.15 (training: validation: test) to prevent under- and overfitting problems of training, validation, and test data in surrogate models. The surrogate model was validated through the correlation coefficient of each dataset. Table 2 shows the configuration of the neural network for constructing the surrogate model. In addition, the sigma and lambda values of the training function were utilized as \(5\times 10^{-5}\) and \(5\times 10^{-7}\), respectively, and the minimum gradient was set to \(1\times 10^{-8}\) for learning parameters. The number of hidden layers and neurons was determined through trial and error, and learning parameters that could be converged by a 48-dimensional function with five variables were considered as representative values. Figure 7 shows that the mean squared error (MSE) values of the training, validation, and test sets of the ANN-based surrogate model converge as the epoch increases. In this study, 30,000 epochs were considered to avoid underfitting and overfitting problems. Figure 8 shows the correlation coefficient (\(R\)) values for the training, validation, and test data of the neural network as 0.9853, 0.9759, and 0.9760, respectively, and the correlation coefficient for 100,000 data was calculated as 0.9825. As a result of the coefficient correlation of the constructed surrogate model, it was confirmed that the accuracy of the network performance response can be predicted with high accuracy. \begin{table}
\begin{tabular}{c c} \hline \hline
**Damage State** & **Reduced Capacity** \\ \hline Complete damage & 0\% \\ Extensive damage & 25\% \\ Moderate damage & 50\% \\ Slight damage & 75\% \\ No damage & 100\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reduced capacities of damaged bridge structure. \begin{table}
\begin{tabular}{c c} \hline \hline
**Network Properties** & **Utilized Function or Value** \\ \hline Network input dimensions & 48(1) \\ (output dimensions) & Feed-forward backpropagation \\ Network type & Scaled conjugate gradient backpropagation \\ Training function & Gradient descent with momentum weight and bias learning function \\ Performance function & Mean squared error \\ Number of layers & 20 \\ Number of neurons & 20 \\ Transfer function & Hyperbolic tangent sigmoid transfer function \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Neural network construction for surrogate model. Figure 8: Correlation coefficients of (**a**) training, (**b**) validation, (**c**) test, and (**d**) all data set.