## 2 The Methodology
### Gated Recurrent Unit Network  
GRU network is the optimized structure of the recurrent neural network (RNN) [25]. However, when the input information is increased to a certain length, the RNN cannot connect to the relevant information. GRU network is aimed at solving the problem of long-range dependence and gradient disappearance of RNN. The GRU neural network with less threshold structure and better efficiency is directly selected for the diagnosis of gear pitting fault. Note that similar to GRU, a recurrent unit in RNN called long short term memory (LSTM) can also be used. Both LSTM and GRU have the same goal of tracking long-term dependencies effectively while mitigating the vanishing/exploding gradient problems. As pointed in [PERSON] et al. [26], after evaluating LSTM and GRU units on the tasks of polyphonic music modeling and speech signal modeling, they found GRU to be comparable to LSTM. Therefore, the GRU is used in this paper as a recurrent unit the same as LSTM. For this reason, it is expected that LSTM will give similar results as GRU. RNNs are widely used in the field of natural language processing. Unlike traditional feedforward neural networks, RNN introduces directional loops that can handle correlated inputs. As so, it can be used to process sequence data. The basic structure of an RNN is shown in Figure 3. In Figure 3, \(\mathbf{x}\), \(\mathbf{h}\), and \(\mathbf{s}\) represent the input, output, and hidden states, respectively. \(\mathbf{U}\), \(\mathbf{V}\), and \(\mathbf{W}\) represent the weight matrix between the input and hidden layers, hidden layers and outputs, and the hidden layers, respectively. The GRU unit specific update process is as follow: First, the two gates in the GRU that control the direction of the data flow are r and z. The update gate model in the GRU neural network is calculated in Equation (4):  
\[z_{t}=\sigma(\mathbf{W_{Z}}\mathbf{h_{t-1}}+\mathbf{U_{z}}\mathbf{x_{t}}+\mathbf{b_{z}}) \tag{4}\]  
In Equation (4), the \(z_{t}\) represents the update gate, \(\mathbf{h_{t-1}}\) represents the output of the previous neuron, \(\mathbf{x_{t}}\) represents the input of the current neuron, \(\mathbf{W_{Z}}\) represents the weight of the update gate, \(\mathbf{U_{Z}}\) represents the weight of the current neuron, and \(\sigma\) represents the sigmoid function. The update gate  
Figure 3: Expansion model of the recurrent neural network. is operated by \(\mathbf{h_{t-1}}\) and \(\mathbf{x_{t}}\), and then it uses the sigmoid function to process. For the update gate \(z_{t}\), when the value is larger, more information in the previous neuron will be retained. If \(z_{t}\) is close to 1, it is equivalent to copying the previous hidden layer information to the current layer. It can learn long distance dependence. The reset gate model in the GRU neural networks is calculated in Equation (5):  
\[r_{t}=\sigma(\mathbf{W_{t}}\mathbf{h_{t-1}}+\mathbf{U_{t}}\mathbf{x_{t}}+\mathbf{b_{r}}) \tag{5}\]  
In Equation (5), \(r_{t}\) represents the reset gate, \(\mathbf{h_{t-1}}\) represents the output of the previous neuron, \(\mathbf{x_{t}}\) represents the input of the current neuron, \(\mathbf{W_{t}}\) represents the weight of the reset gate, \(\mathbf{U_{t}}\) represents the weight of the current neuron, and \(\sigma\) represents the sigmoid function. The reset gate r is operated by \(\mathbf{h_{t-1}}\) and \(\mathbf{x_{t}}\), and then it uses the sigmoid function to process. For the reset gate, when its value is 0, it means to discard the information from the previous neuron.