# Fault Diagnosis of Planetary Gear Train Crack Based on DC-DRSN
## 2 Theory and Method
### Neural Network Model  
DRSN is essentially a network integrated by ResNet with an attention mechanism and a soft threshold function. The core of ResNet is the residual block unit (RBU), as shown in Figure 1, which uses multiple parametric layers to learn the residual representation between the input and output. The input feature \(x\) is first stored in the variable, and then the residual part \(F(x)\) is learned through a series of nonlinear layers, and finally the residual is calculated. Through cross-layer identity mapping, RBU transforms the learning objective from the mapping between input and output to learning residuals, which reduces the training difficulty and accelerates the network convergence speed. It is expressed by the formula [16]:  
\[H(x)=F(x)+x \tag{8}\]The residual shrinkage block unit (RSBU) is the core of the DRSN network, and the structure is shown in Figure 2, which introduces the attention mechanism and soft thresholding on the basis of RBU. The attention mechanism uses the squeeze-and-excitation network (SENet) proposed by [PERSON] [27], which learns the weights of each channel through the small network \"global pooling \(\rightarrow\) fully connected layer \(\rightarrow\) ReLu activation function \(\rightarrow\) fully connected layer \(\rightarrow\) Sigmoid function\", and applies it to the original input feature map, and weights the eigenvalues of each channel through multiplication operations to adjust the feature size of each channel. In this way, the attention of the network to important features is enhanced so as to improve the expressiveness, generalization ability, and robustness of the model. In this paper, the soft thresholding of different thresholds of channels is adopted, which can automatically calculate different scales and thresholds for each channel, so that the features of different channels are subject to different constraints in the sparsity process, allowing the network to process the features more flexibly. If \(\alpha_{c}\) is the threshold of the \(c\)th channel of the feature map, and \(\underset{i,j}{average}|x_{i,j,c}|\) is the average value of the absolute value of each channel of the input feature \(x\), the threshold is calculated as follows [17]:  
\[\tau_{c}=\alpha_{c}\cdot\underset{i,j}{average}|x_{i,j,c}| \tag{9}\]  
Figure 1: The residual module of the deep residual network.  
Figure 2: The RSBU with different thresholds of channels.  
Soft thresholding is the core step of signal denoising, which is combined with deep learning to decompose the input signal through the convolutional layer, then filter all the signals after decomposition within the threshold, and then reconstruct the filtered signal to achieve the purpose of denoising. The soft threshold function is as follows. After deriving the soft threshold function, the gradient is only 0 or 1, which avoids gradient explosion and gradient disappearance.  
\[y=\begin{cases}x-\tau&x>\tau\\ 0&-\tau\leq x\leq\tau\\ x+\tau&x<-\tau\end{cases} \tag{10}\]  
where \(x\), \(y\), and \(\tau\) are the input, output, and threshold values, respectively.