## 2 Proposed Method
### Proposed ANN-Based Surrogate Model  
In this study, TSTT was introduced as the performance index of a bridge transportation network, and an ANN-based surrogate model with the MCS framework was proposed to predict TSTT. Figure 2 shows a flowchart for integrating a surrogate model with the MCS framework to evaluate the mean TSTT and relative importance measure. The input data required to execute system reliability are the topology data (nodes, links, adjacent matrix) of the bridge transportation network, spatially correlated seismic attenuation law of the target network, and damage states and fragility curve parameters of the bridge components. The United States Federal Emergency Management Agency (FEMA) [37] proposed the failure probability of a bridge structure as five levels of damage states and estimated the fragility curve based on past empirical data as a lognormal curve (expressed as a median and a log standard deviation). A fragility curve is defined as the conditional probability based on the ground intensity measure [38], and FEMA determined that spectral acceleration (SA) predicts the failure probability of the bridge structure well. In the case of the seismic attenuation law, the ground motion prediction equation proposed by [PERSON] et al. [39] was utilized for non-linear regression analysis of 222 earthquake datasets from 132 stations in South Korea. The intensity of the mean ground motion from that study can be expressed by the following equation:  
\[ln\big{(}SA_{ij}\big{)}=c_{1}+c_{2}M_{i}+c_{3}ln\bigg{(}\sqrt{{R_{ij}}^{2}+h^ {2}}\bigg{)}+c_{4}R_{ij}+c_{5}s \tag{1}\]where \(SA_{ij}\) represents the spectral acceleration at site \(j\) when an earthquake occurs at source \(i\), \(M_{i}\) represents the earthquake magnitude at the epicenter \(i\), \(R_{ij}\) is the epicentral distance between source \(i\) and site \(j\), \(h\) is the length of focal depth (km), \(c_{1}\)-\(c_{5}\) represent non-linear regression coefficients (in this study, the regression coefficients are assumed to be \(-\)5.15, 0.95, \(-\)0.92, 6.8, \(-\)0.0003, and 0.208, respectively [39]), and \(s\) denotes a dummy variable and is assumed to be either \(-\)1, 0, or 1 depending on the station. In addition, in this study, inter- and intra-event terms were considered to represent the uncertainty of ground motion [40,41]. Inter-event terms indicate the uncertainty with which the energy transmitted to the ground surface varies owing to the characteristic of the earthquake itself, whereas intra-event terms indicate the uncertainty with which the transmitted energy to the ground surface varies depending on geotechnical environments and seismic attenuation paths. The uncertainty of ground motion intensity \(\rho_{total}\) can be expressed by the following equation [42,43]:  
\[\rho_{total}=\frac{{\sigma_{\eta}}^{2}}{{\sigma_{\eta}}^{2}+{\sigma_{\epsilon }}^{2}}+\frac{{\sigma_{\epsilon}}^{2}}{{\sigma_{\eta}}^{2}+{\sigma_{\epsilon} }^{2}}\rho\Big{(}\Delta_{ij}\Big{)} \tag{2}\]  
where \(\sigma_{\eta}\) and \(\sigma_{\epsilon}\) represent predefined inter- and intra-event residuals with zero mean value, and \(\rho\Big{(}\Delta_{ij}\Big{)}\) represents the spatial correlation equation. In this study, the intra-event correlation equation proposed by [PERSON] and [PERSON] [44] was utilized, and the correlation function can be expressed by the following equation:  
\[\rho\Big{(}\Delta_{ij}\Big{)}=e^{(-0.509\sqrt{\Delta})} \tag{3}\]  
where \(\Delta_{ij}\) represents the distance between the \(i\)th site and the \(j\)th site. Once the input data required to apply the ANN-based surrogate model with the MCS framework are acquired, the epicenter and magnitude of the earthquake are determined according to each scenario. Figure 2: Flowchart for accelerated seismic risk assessment of bridge transportation networks. Depending on the epicenter and the earthquake magnitude, the failure probability of each bridge component can be evaluated according to the source-to-site distance.