## 2 Methodology
### Benchmark Model  
The LSTM model and the GRU model are both variants of RNN, which both excel in handling time series problems, and therefore, are considered the benchmark models for this study. LSTM addresses the issues of vanishing and exploding gradients in RNN by selectively remembering information through input gates, output gates, forget gates, and memory cells [37; 38], as shown in Figure 1a. Specifically, when data enters an LSTM unit, it first encounters the input gate \(\mathbf{i}_{t}\) and the forget gate \(\mathbf{f}_{t}\). The input gate decides how much new input information should be incorporated into the cell state \(\mathbf{c}_{t}\), while the forget gate determines how much of the old information should be retained or discarded. The operations of these gates are based on the current input \(\mathbf{x}_{t}\), the previous hidden state \(\mathbf{h}_{t-1}\), and the previous cell state \(\mathbf{c}_{t-1}\). Then, the cell state \(\mathbf{c}_{t}\) is updated, combining new information and forgotten old information, as determined by the input and forget gates. Finally, the output gate \(\mathbf{o}_{t}\), based on the updated cell state and the previous hidden state, decides the final output \(\mathbf{h}_{t}\), which is used for the next time step or LSTM unit.  
\[\mathbf{i}_{t}=\sigma(\mathbf{W}_{\text{xi}}\mathbf{x}_{t}+\mathbf{W}_{\text{hi}}\mathbf{h}_{t-1}+ \mathbf{W}_{\text{ci}}\odot\mathbf{c}_{t-1}+\mathbf{b}_{\text{i}}) \tag{1}\]  
\[\mathbf{f}_{t}=\sigma(\mathbf{W}_{\text{xf}}\mathbf{x}_{t}+\mathbf{W}_{\text{hi}}\mathbf{h}_{t-1}+ \mathbf{W}_{\text{cf}}\odot\mathbf{C}_{t-1}+\mathbf{b}_{t}) \tag{2}\]  
\[\mathbf{\tilde{C}}_{t}=\tanh(\mathbf{W}_{\text{xc}}\mathbf{x}_{t}+\mathbf{W}_{\text{hc}}\mathbf{h} _{t-1}+\mathbf{b}_{\text{c}}) \tag{3}\]  
\[\mathbf{C}_{t}=\mathbf{f}_{t}\odot\mathbf{C}_{t-1}+\mathbf{i}_{t}\odot\mathbf{\tilde{C}}_{t} \tag{4}\]  
\[\mathbf{o}_{t}=\sigma(\mathbf{W}_{\text{xo}}\mathbf{x}_{t}+\mathbf{W}_{\text{ho}}\mathbf{h}_{t-1}+ \mathbf{W}_{\text{co}}\odot\mathbf{C}_{t}+\mathbf{b}_{\text{o}}) \tag{5}\]  
\[\mathbf{h}_{t}=\mathbf{o}_{t}\odot\tanh(\mathbf{C}_{t}) \tag{6}\]  
where \(\sigma\) denotes the sigmoid function and tanh represents the hyperbolic tangent function; \(\mathbf{x}_{t}\), \(\mathbf{h}_{t}\), and \(\mathbf{C}_{t}\) represent the current input, hidden state, and cell state, respectively; \(\odot\) represents the Hadamard product; \(\mathbf{W}_{\text{xi}}\), \(\mathbf{W}_{\text{xf}}\), \(\mathbf{W}_{\text{xc}}\), \(\mathbf{W}_{\text{xc}}\), \(\mathbf{W}_{\text{xo}}\), \(\mathbf{W}_{\text{cf}}\), \(\mathbf{W}_{\text{co}}\), \(\mathbf{W}_{\text{hi}}\), \(\mathbf{W}_{\text{hf}}\), \(\mathbf{W}_{\text{hc}}\), and \(\mathbf{W}_{\text{ho}}\) represent the weight matrices; and \(\mathbf{b}_{\text{i}}\), \(\mathbf{b}_{\text{f}}\), \(\mathbf{b}_{\text{c}}\), and \(\mathbf{b}_{\text{o}}\) denote the bias terms. Based on LSTM, GRU simplifies the forget gate and output gate into an update gate \(\mathbf{z}_{t}\), and combines the cell state with the hidden layer into a reset gate \(\mathbf{r}_{t}\)[39], as shown in Figure 1b, where \(\mathbf{\tilde{C}}_{t}\) represents the candidate vector, and \"1-\" signifies the operation of subtracting the output of the update gate \(\mathbf{z}_{t}\) from 1, which is used to calculate the proportion of old information in the new hidden state.