## 2 Methodology
### Application of the BP-ANN
Equation (10) demonstrated that, initially, the BP-ANN should be suitable for simulating the transfer matrix \(T\) without considering any time-related factors, despite the reconstruction of this study being for the purpose of predicting time-series signals. Moreover, once a network is trained to simulate the transfer matrix \(T\), it should be applicable to various scenarios, including those with fluctuating temperature and traffic conditions. Additionally, this study did not delve into modifications or sophisticated algorithms for setting hyperparameters. Moreover, for certain typical prestressed bridges, the moment of inertia of the cross-section may vary along the length of the bridge due to the positioning of the prestressing cables, suggesting that the mode shape function would differ from that of equivalent non-prestressed bridges. Nevertheless, mode shape information was inherent and incorporated during the training process. As a result, the trained strain-related transfer matrix, which was obtained from monitoring data, differed in accordance with the arrangement of the prestressing cables. Upon completion of the bridge's construction, both the mode shape function and the strain-related transfer function should be established, and they should remain time-independent and constant. Consequently, the proposed method should also be applicable for prestressed bridges. Furthermore, as indicated from Equations (8a) and (8b) to (10), the mode shapes that form the output signal must be encompassed within those present in the input signal. Essentially, the mapping is the ratio of the mode coordinates across different cross-sections. For the data in the output neuron, if a mode shape's modal coordinate is non-zero, corresponding to \(-sin\frac{n\pi\pi}{L}\) in Equation (5b), one of the input features should also include the non-zero coordinate of that mode shape. Failing to do so would lead the neural network to attempt an orthogonal mapping between the input and output signals, which is assumed to be an erroneous mapping. For example, in the case of a simply supported beam, the modal coordinate of the second mode shape is zero. Calculating the mapping of the second mode coordinate between the mid-span and other cross-sections is problematic due to the transition from zero to non-zero values. To address this issue, it is recommended to analyze the frequency spectra of the high-frequency responses across all cross-sections. Randomly selecting inputs, meaning responses at arbitrary cross-sections, may not result in physically meaningful outcomes. Additionally, for data processing, the high-frequency bridge responses were isolated from the raw monitoring data using a high-pass filter. For short-span bridges, with lengths  
Figure 3: The ANN structure used in this study to simulate the intersectional relationship (C-S- denotes the monitoring data at the different cross-sections and C-S-_Ronstrc_ denotes the responses to be reconstructed). around 30 m, the first natural vibrational frequency generally exceeds 3 Hz [38]. For bridges with spans ranging from 30 to 60 m, the primary bridge frequency consistently surpasses 2 Hz. In light of these considerations, the cutoff frequency for the high-pass filter in this study was established at 1 Hz.