# Structural Damage Detection Based on One-Dimensional Convolutional Neural Network
## 2 Formulation of Analytical Model
### Convolutional Neural Networks  
A typical convolutional neural network consists of a convolutional layer, a pooling layer, and a fully connected layer. These layers work together to extract features from the data. Then, the classification is completed by the fully connected layer. Through the series of operations, the convolutional neural network can reduce the dimensionality of data and make it easier to be trained. The details of the proposed convolutional neural network architecture are shown in Table 2.  
(1) Convolutional layer  
A traditional neural network consists of an input layer, a hidden layer, and an output layer. There are many neurons in each layer. Additionally, the neurons between two adjacent layers are fully connected. Therefore, it produces a large number of parameters. Excessive parameters will cause gradient dispersion. Convolution can reduce parameters in the neural network and improve learning efficiency. Therefore, the convolutional layer is the most important component of the convolutional neural network. Two-dimensional convolution is often used in image processing. However, the input data are interlayer displacement in this paper, and one-dimensional convolution is very suitable for this kind of damage detection. The data is inputted into the convolutional layer, and the convolutional kernel in the convolutional layer and the input data are convoluted. The input shape of the first convolutional layer is 50 \(\times\) 12, the kernel size is 5 \(\times\) 12, the kernel number is 32. Each convolution operation only processes a part of the input data. The convolutional kernel starts at the upper left corner of the input data, and the parameters of the convolutional kernel are multiplied by the parameters of the coverage area. Then, the multiplied values are summed. The convolutional kernel moves one element (stride = 1) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & \multirow{2}{*}{**Input Shape**} & **Output** & **Kernel** & \multirow{2}{*}{**Kernel Size**} & \multirow{2}{*}{**Activation**} \\  & & **Shape** & **Number** & & \\ \hline Convolution 1-D & (50, 12) & (46, 32) & 32 & \(5\times 12\) & ReLU \\ Max Pooling 1-D & (46, 32) & (23, 32) & None & \(2\times 32\) & None \\ Convolution 1-D & (23, 32) & (19, 64) & 64 & \(5\times 32\) & ReLU \\ Max Pooling 1-D & (19, 64) & (9, 64) & None & 2 & None \\ Dropout & (9, 64) & (9, 64) & None & None & None \\ Flatten & (9, 64) & (576) & None & None & None \\ Dense & (576) & (512) & None & None & ReLU \\ Dense & (512) & (4) & None & None & Softmax \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The details of the proposed CNN architecture. Figure 3: Sample generation. down for the next convolution. The initial parameters within the convolutional kernel are randomly generated. In this convolutional neural network, each convolutional kernel can generate a 46 \(\times\) 1 matrix with input data. Thus, the output shape of the first convolutional layer is 46 \(\times\) 32. The convolution operation diagram is show in Figure 4.  
(2) Activation function  
In order to simulate the human nervous system, a neural network is proposed. The human nervous system responds only to some neurons. The activation function is also intended to activate some neurons. The activation of neurons is determined by the activation function. The activation function is divided into linear activation function and nonlinear activation function. Linear activation functions can only provide linear mapping capabilities and do not improve the expressive power of the network. Therefore, the nonlinear activation function has been widely used. Common nonlinear activation functions are sigmoid, tanh, and rectified linear unit (ReLU). These nonlinear activation functions are shown in Figure 5.  
As can be seen from Figure 5, the derivatives of sigmoid and tanh are zero when x approaches infinity. This situation easily causes the vanishing gradient problem, and the state of network cannot be updated.