## 2 Theoretical Analysis
### Respondruction with BP-ANN  
To model the transfer matrix described in Equation (7), which specifically maps the responses from \(x_{1}\) and \(x_{2}\) to \(x_{3}\), we have employed a BP-ANN algorithm for training. The BP-ANN is configured with a time-invariant architecture, which is consistent with the time-independent relationship detailed in Equation (7). It is important to note that the BP-ANN serves as a pure simulation of the aforementioned transfer matrix, rendering an explicit transfer matrix unnecessary. The primary purpose of using BP-ANN is to mitigate the potential interference from measurement noise, rather than to enhance the sensitivity of the proposed method. As introduced in Section 2.2, the sensitivities of the proposed method and the utilized damage index are inherently aligned with the measured physical index. During the network training phase, data from multiple cross-sections are configured as input features [33], with the response at a specific cross-section designated as the output. The network is then trained to establish a mapping that reflects the inter-sectional dynamics. This well-trained network is engineered to grasp the inter-sectional relationship amidst varying traffic conditions. Remarkably, it can adeptly utilize these dynamics without needing explicit traffic data, harnessing the intrinsic structural relationships within the bridge, which is pivotal for online monitoring. The BP-ANN is a widely utilized machine learning technique. Details regarding the construction of the network's basic units can be found in existing research [34]. The network architecture employed in this study is illustrated in Figure 2[27]. For example, consider a scenario where responses at \(i\) cross-sections are measured. The time-series responses at the first m cross-sections are used as m inputs, while the response at the \(i\)-th cross-section is set as the output. Specifically, for each set of inputs and output, each neuron contains a single time-point value. The network excludes time-dependent information related to traffic conditions. Moreover, the training process of our BP-ANN does not necessitate conventional feature selection. As delineated in Section 2.1, our study is confined to the first two vibration modes. Consequently, we employed responses from two cross-sections, effectively using two features as inputs. Rather than the typical approach in existing studies that seeks to identify the most correlated input feature among numerous options, we utilized the measured responses from both cross-sections. This approach underscores the advantage of our proposed data-driven method. In contrast to finite element model-based methods, the criterion for feature selection in this study is that the mode shapes represented in all inputs must include the mode shape component present in the output. If there is a unique mode in the output that is orthogonal to all modes in the input, as referenced in [27], the model may fail to yield satisfactory results. This consideration is pivotal for ensuring the network's proficiency in capturing the dynamics across sections and accurately reconstructing responses for damage localization. However, such a coincidence is unlikely to occur frequently, which facilitates the practical application of using a minimal number of sensors--specifically, two sensors at two distinct cross-sections--to achieve data reconstruction. Furthermore, instead of selecting the most correlated features as referenced in [35], a critical consideration in practical applications is the selection of regions that are less susceptible to damage. When a bridge sustains damage, stress concentration or fatigue failure might manifest in certain areas, leading to nonlinear phenomena. Installing a sensor in such regions could result in a nonlinear relationship between the responses measured at this sensor and those at other sensors where no damage is present, which would not meet the application requirements of our proposed method.