## 3 Inversion Method for the Dynamic Modulus of Pavement Structures  
### Convolutional Neural Network  
A CNN can effectively reduce the number of network parameters, alleviate the model overfitting problem, and reduce the memory occupied by deep networks. A standard fitting CNN is usually composed of convolution layers, pooling layers, activation layers, fully connected layers, and regression layers. In practical applications, the convolution and activation layers are often referred to together as convolution layers. The data input (network input) transmits through a series of layers, and is finally mapped to the fitting value corresponding to the original data through regression layers. In particular, the input of a 1-D CNN is a 1 \(\times\)\(N\) or \(N\)\(\times\) 1 matrix. Figure 6 presents the basic structure of the CNN. The feature extraction is carried out through the convolution layer and the pooling layer, and the fitting value (parameter value of soil layers 1-3) is finally obtained in the regression layer.  
Figure 5: The response of the generic multi-layered pavement system [27]. **(a)** Displacement-time curve at deflection points; **(b)** Maximum displacement curve at deflection points.  
Figure 6: The basic structure of the CNN.  
A 1-D CNN is often used in sequence models. Figure 7a presents a typical 1-D CNN operation, in which the original data of the convolution layer are an input matrix whose size is 1 \(\times\) 6, and the convolution kernel is a matrix whose size is 1 \(\times\) 3. The convolution operation process is to multiply each element in the convolution kernel with the corresponding element in a sub-region (e.g., orange box or purple dotted box) of the original data of the convolution layers, and then add the results to obtain an element in the feature vector. Each time the convolution operation is performed, the convolution kernel moves down with a certain step size (the step size is 1 in Figure 7a), and the operation process is repeated until all elements of the original data are traversed. Finally, after the convolution operation, a new array is formed (for example, the feature vector [8 23 6 29]).  
The pooling operation is a down sampling technique. By reducing the characteristic dimensions of the output after convolution, the network parameters can be effectively reduced, thus accelerating the calculation process. This can greatly improve the computational speed of the CNN and effectively prevent overfitting. During pooling, the input matrix is divided into several regions, and the maximum value is output for each sub-region. There are usually two different pooling methods, namely maximum pooling and average pooling. For the purposes of this study, maximum pooling is superior to average pooling. Figure 7b shows the maximum pooling operation.  
The activation function introduces nonlinear factors into the neural network and improves its fitting ability to learn complex features. Thus, the neural network can approximate any nonlinear function, and the convergence of the network can also be accelerated. Common activation functions include the Sigmoid, Tanh, ReLU, and Leaky ReLU. The activation function used in this study is Leaky ReLU, which is similar to ReLU and helps to expand the applicability of the ReLU function. This function retains the characteristic information that the negative values may contain. It has greater application significance than the ReLU function. Its form is  
\[\text{Leaky ReLU}(x)=\begin{cases}x,\ x\geq 0\\ \alpha x,x<0\end{cases} \tag{20}\]  
where \(x\) is the input value of the network, and \(\alpha\) is 0.01 by default.  
Figure 7: The basic principle of the convolution and pooling operations.