## 5 Calibration of Structural Capacity
### Prediction of Response Ratio Using ANN  
In this section, an artificial neural network (ANN) model that can predict the response ratio based on the bridge information and condition evaluation results in the diagnosis report was developed.  
An ANN is based on a feed-forward backpropagation (BPP) algorithm, so the predicted value reaches the target value. The ANN model consists of an input layer, a hidden layer, and an output layer, and each layer is connected by weights. The provided input values are propagated to the output layer through the hidden layer, and the calculated result value is displayed in the output layer. This process is called a forward propagation process, and the resulting value output through forward propagation represents the error from the provided target value to evaluate the model's performance. At this time, the calculated error is propagated in the reverse direction, and the weights connected between each layer are adjusted. This process is called backpropagation, and the forward propagation and backpropagation processes are repeated, so the error converges to zero.  
In order to model an optimal ANN, which is essential, various hyper-parameters, such as a training algorithm, an optimizer, a performance function, the number of hidden layers, and the number of nodes in the hidden layers, can be used [25]. According to numerous previous studies using ANNs, the training algorithm in the BPP learning process shows a low level of error and a level of high precision when using the Levenberg-Marquardt (LM) algorithm [26; 27; 28; 29]. Thus, in this study, the ANN model was trained using the LM algorithm, and the gradient descent method (GDM) was adopted as an optimization function to increase the training efficiency. The default setting of the training used by the LM algorithm in MATLAB is that the data are divided as follows: training (70%), validation (15%), and testing (15%). For a better study of the insufficient data, k-fold cross-validation is performed to evaluate model errors realistically and to prevent overfitting [25]. In this study, five-fold cross-validation was used to train 82 sets of data. Additionally, the mean squared error and sum of squares error functions were used as the criteria for stopping the network training process, and an appropriate performance evaluation function was determined by a trial-and-error method.  
When the BPP neural network is applied, the structure of the hidden layer largely determines the training speed and generalization ability [30]. Increasing the number of hidden layers is advantageous when desiring ANNs with a higher accuracy, but sometimes this causes overfitting problems [31]. Simulations according to the number of hidden layers have been performed by many scholars, which proved that the BPP neural network has a sufficient generalization ability with only a single hidden layer, when there are a sufficient number of neurons in the hidden layer [31; 32; 33]. Accordingly, the current ANN model was also developed into a structure using a single hidden layer. Then, a trial-and-error process was used, so the optimal number of neurons could be specified using the trial-and-error method. Figure 9 shows a schematic diagram of the proposed neural network.