## 2 Methodology
### Application of the BP-ANN  
Section 2.1. demonstrated that the target relationship is constant and time-invariant. Consequently, this study employed the BP-ANN [37] to model this relationship. A neural network [37] is composed of an input layer, several hidden layers, and an output layer, interconnected through neurons. Each neuron in the network processes multiple inputs and generates an output signal via a transfer function, commonly referred to as an activation function. Connections between neurons are quantified by weights. The standard architecture of a neural network is depicted in Figure 2, featuring three distinct layers: the input layer (I), the hidden layer (H), and the output layer (O). Specifically, each neuron in the hidden layer integrates an activation signal, which is the aggregate of all incoming inputs weighted accordingly [37], and subsequently generates an output via a transfer function:  
\[x_{j}=\sum_{i}l_{i}W_{ij} \tag{16}\]  
where \(x_{j}\) denotes the activation signal at the \(j\)-th neuron in the hidden layer, \(l_{i}\) is the input of the \(i\)-th neuron in the input layer, and \(W_{ij}\) is the corresponding weight factor from \(l_{i}\) to \(x_{j}\). The neurons in the output layer receive the following activation signals from the hidden neurons [37]:  
\[y_{j}=\sum_{i}h_{i}W_{jk} \tag{17}\]  
where \(h_{i}\) is the output of the \(j\)-th neuron in the hidden layer, \(W_{jk}\) is the weight of the connection between the neurons \(j\) and \(k\) in the hidden and output layers, respectively, and \(y_{j}\) denotes the input to the \(j\)-th neuron in the output layer. These inputs \(y_{j}\) (\(j=1,2\),..) are transformed again to give the outputs of the neural network [37]:  
\[o_{k}=\frac{1}{1+e^{-y_{k}}} \tag{18}\]  
in which \(o_{k}\) are predicted values of the network. The error function at the output neurons is defined as  
\[\mathrm{E}(W)=\frac{1}{2}{\sum_{k}}{(d_{k}-o_{k})}^{2} \tag{19}\]  
where \(d_{k}\) and \(o_{k}\) represent the desired and predicted values of the outputs, respectively. The error function should be minimized so that the neural network achieves the best performance. Numerous algorithms have been crafted to reduce the error function \(\mathrm{E}(W)\). The backpropagation (BP) algorithm stands out as the most prevalent, functioning as a supervised learning method. Within a BP neural network, the output layer's error is retroactively disseminated through the hidden layer back to the input layer, aiming to refine the final output. The network's weights are iteratively calculated and refined using the gradient descent approach to diminish the output error. Throughout the training phase, the network internalizes the correlations between system inputs and outputs via the interplay of connection weights. This study adopted the foundational approach of the BP-ANN, as previously outlined in [37]. As depicted in Figure 3, responses measured at several cross-sections on the bridge served as inputs (i.e., \(l_{i}\) in the input layer in Figure 2), and the response at a particular cross-section (i.e., \(o_{k}\) illustrated in Equation (17)) constituted the output. The network is trained utilizing historical monitoring data. In this study, the BP-ANN was trained with data collected from field tests. Additional specifics on the BP-ANN configuration are provided in Section 4.2. It is important to note that the network inherently accounts for the  
Figure 2: Structure of a typical ANN [37]. potential impacts of unknown traffic conditions and air temperature on high-frequency responses, as these factors are encapsulated within the input-output mappings without any need for explicit adjustment. The aforementioned derived physical characteristics do not modify the BP-ANN structure; they merely determine the selection of the proposed method and the necessity for the inputs and outputs of the network.