## 3 Method
### ResNet  
Convolutional Neural Networks (CNNs) have long been a significant class of models in the field of machine learning. Since [PERSON] et al. introduced the method of extracting feature maps from data using convolutional kernels [41], CNNs have been widely applied in image recognition, object classification, and natural language processing [42; 43; 44]. With in-depth research in the field of machine learning and the support of computer hardware devices, researchers have attempted to use CNNs with more complex structures and greater depths. However, simply increasing the depth of a CNN does not necessarily enhance model performance [45]; on the contrary, overly complex structures and a large number of parameters can sometimes affect the training efficiency of the model. Consequently, researchers have begun to focus on whether newly added layers (blocks) effectively improve the performance of neural networks. Building on the rationale, [PERSON] introduced ResNet [46]. In general, when the depth of a CNN reaches a certain level, the training error no longer decreases but instead increases, a phenomenon we refer to as vanishing or exploding gradients. Within CNN, feature matrices are processed through convolutional layers to extract feature information and then passed through activation functions to enhance the model's nonlinear expression capabilities. The process of repeatedly applying convolutions completes the training. For a network composed of \(N\) layers, the first \(N-1\) layers can be described as follows:  
\[f^{1}(I;W_{1},b_{1})=\sigma(W_{1}*I+b_{1}) \tag{4}\]  
\[f^{n}(I;W_{1},b_{1},\ldots,W_{n},b_{n})=\sigma\Big{(}W_{n}*f^{n-1}(I;W_{1},b_ {1},\ldots,W_{n-1},b_{n-1})+b_{n}\Big{)} \tag{5}\]  
where \(f\) denotes the convolution operation; \(W_{i}\) and \(b_{i}\) represent the weight and bias parameters of the \(n\)-th layer, respectively; \(I\) denotes the input data; and \(\sigma\) is the activation function. The specific structure is illustrated in Figure 5a. It can be observed that the data on this layer are derived from the convolution of the data from the previous layer. When an inappropriate weight or activation function is used, the continuous propagation through multiple convolutional layers may lead to a degradation problem in the network (such as vanishing and exploding gradients). Figure 4: Sub-pixel convolution process. Residual structures were introduced to address the issue of network degradation that arises with the increase in model depth. By designing residual blocks to establish skip connections between feature matrices of different layers, the problem of inappropriate weight parameters being amplified during transmission is reduced. This can be articulated as follows:  
\[f^{n}(I;W_{1},b_{1},\dots,W_{n},b_{n})=\sigma\Big{(}W_{n}*f^{n-1}(I;W_{1},b_{1}, \dots,W_{n-1},b_{n-1})+b_{n}\Big{)}+F(I) \tag{6}\]  
where \(F\) denotes a residual block. Traditional CNNs have attempted to model the nonlinear mapping \(f(I)=I\) between the input data \(I\) and the corresponding output \(f(I)\) through the stacking of multiple convolutional layers. During the actual training process, this identity mapping is challenging to learn, and the pursuit of this mapping through an excessive number of convolutional layers often leads to the network degradation problem. ResNet does not directly learn this mapping but instead focuses on the difference between the input and output, achieving an identity mapping through alternative structural means [46]. In ResNet, the problem is reformulated as \(f(I)=I+F(I)\), where the identity mapping is explicitly integrated into the network architecture. This transformation shifts the focus to learning the residual \(F(I)=f(I)-I\), which simplifies the optimization process and enhances the network's ability to model complex functions [47]. This approach makes it easier to find the mapping function compared to traditional CNNs. Moreover, it achieves the same effect with a model that is less deep and less complex, as shown in Figure 5b.