## 3 Methodology
### Using the Recursive Layer to Capture Inference Features  
In addition, the recursive layer is utilized to better extract feature information by capturing inference features. In this recursive layer, four structures can be chosen for information capture, including MLP, RNN, LSTM, and GRU. In detail, these structures can be represented as follows:  
1. Multi-layer perceptron. MLP is a feed-forward network that learns weights \(\theta\), mapping inputs to outputs. For a model with two hidden layers, the chain structure of multiple stacked layers provides depth. In order to improve the learning ability of the model, nonlinear functions, such as \(ReLU\) or \(sigmoid\) functions, are applied to the neuronal output. The optimal weight is determined by minimizing a differentiable loss function using backpropagation, which updates the network weight by propagating the gradient of the weight relative to the loss function back through the network. 2. Recurrent neural network. The main purpose of the RNN, a structural diagram of which is shown in Figure 5, is to process and predict sequence data. The calculation process of the classic RNN can be expressed as Formulas (22) and (23):  
\[h_{t}=f(Ux_{t}+Wh_{t-1}+b) \tag{22}\]  
\[y=softmax(Vh_{t}+c) \tag{23}\]  
where the inputs are \(x_{1}\), \(x_{2}\), \(\cdots\), \(x_{t}\), the corresponding hidden states are \(h_{1}\), \(h_{2}\), \(\cdots\), \(h_{t}\), and the outputs denote \(y_{1}\), \(y_{2}\), \(\cdots\), \(y_{t}\). \(U\), \(W\), and \(V\) denote the weight parameters, \(b\) and \(c\) represent biases, and \(f\) denotes the activation function, which is usually the \(tanh\) function, as follows:Long short-term memory. LSTM, a structural diagram of which is shown in Figure 6, is composed of forget, input, and output gates. These jointly control the storage and deletion of data, as shown in Equations (24)-(28):  
\[f_{t} =\delta(W_{fh}\times h_{t-1}+W_{fx}\times x_{t}+b_{f}) \tag{24}\] \[i_{t} =\delta(W_{ih}\times h_{t-1}+W_{ix}\times x_{t}+b_{i})\] (25) \[o_{t} =\delta(W_{oh}\times h_{t-1}+W_{ox}\times x_{t}+b_{o})\] (26) \[c_{t} =f_{c}\times c_{t-1}+i_{t}\times tanh(W_{o}\times[h_{t-1},x_{t}]+b _{o})\] (27) \[h_{t} =o_{t}\times tanh(c_{t}) \tag{28}\]  
where \(f_{t}\) denotes the forget gate, \(i_{t}\) represents the input gate, and \(o_{t}\) indicates the output gate. \(W_{fh}\), \(W_{fx}\), \(W_{ih}\), \(W_{xh}\), \(W_{oh}\), and \(W_{ox}\) are the related weight matrices. \(b_{f}\), \(b_{i}\), and \(b_{o}\) are the related bias; \(c_{t}\) denotes the data retained from the beginning to the current moment; \(h_{t-1}\) saves the information of the previous time step \(t-1\), while \(h_{t}\) controls how much retained data can be transferred to the next moment. 4 Gated recurrent unit  
As shown in Figure 7, a GRU has only two gates. It combines the input and forget gates found in LSTM into a single gate; this is called the update gate, as depicted in Equations (29)-(32):  
\[z_{t} =\delta(W_{zh}\times h_{t-1}+W_{zx}\times x_{t}+b_{z}) \tag{29}\] \[r_{t} =\delta(W_{rh}\times h_{t-1}+W_{rx}\times x_{t}+b_{r})\] (30) \[h_{t}^{\prime} =tanh(W_{xh}\times x_{t}+r_{t}\times W_{hh}\times h_{t-1})\] (31) \[y=softmax(Vh_{t}+c) \tag{23}\]0  
\[y=softmax(Vh_{t}+c) \tag{23}\]1 Diagram of the structure of an RNN. \[y=softmax(Vh_{t}+c) \tag{23}\]2 Diagram of the structure of an LSTM. where \(z_{t}\) denotes the update gate and \(r_{t}\) represents the reset gate. \(W_{2h}\), \(W_{rh}\), and \(W_{xh}\) denote the related weight matrices. \(b_{z}\) and \(b_{r}\) indicate the related biases. \(h_{t}^{r}\) denotes new information and \(h_{t}\) denotes the output. These layers are replaced with \(RL\). Therefore, the feature vectors extracted by this layer can be represented as shown in Formula (33):  
\[y=softmax(Vh_{t}+c) \tag{23}\]3  
where \(X_{EDEL}^{-}\) represents the feature vector extracted by the GRU layer, and \(RL\) denotes the nonlinear activation function that replaces it.