## 2 Materials and Methods  
### Bidirectional Long Short-Term Memory Network (BiLSTM) Model  
BiLSTM is a deep learning model based on Recurrent Neural Networks (RNNs) that is widely applied in temporal data modeling. The architecture of LSTM was specifically designed to address the vanishing gradient and exploding gradient problems that commonly affect traditional RNNs when processing long-term dependencies [33]. By incorporating gating mechanisms (input gate, forget gate, and output gate), LSTM dynamically regulates information flow pathways, enabling the network to selectively retain or discard temporal information, thereby enhancing its capacity for modeling long-term sequence features. The structure of LSTM is illustrated in Figure 1, with its computational process represented by the following equations:\[\left\{ \begin{array}{l}f_{t}=\sigma(W_{f}[h_{t-1},x_{t}]+b_{f})\\ i_{t}=\sigma(W_{i}[h_{t-1},x_{t}]+b_{i})\\ \widetilde{C}_{t}=\tanh(W_{c}[h_{t-1},x_{t}]+b_{c})\\ C_{t}=f_{t}*C_{t-1}+i_{t}*\widetilde{C}_{t}\\ o_{t}=\sigma(W_{o}[h_{t-1},x_{t}]+b_{o})\\ h_{t}=o_{t}*\tanh(C_{t})\end{array} \right. \tag{1}\]  
where \(f_{t}\) represents the forget gate, \(i_{t}\) represents the input gate, \(o_{t}\) represents the output gate, \(C_{t}\) and \(h\) represent the cell memory and the hidden state, respectively, \(W\) and \(b\) represent the weights and biases of the network, and \(\sigma\) and tanh represent the activation functions. Although LSTM mitigates the vanishing gradient problem through its gating mechanism, it may still face the challenge of information attenuation when processing ultra-long time series data. Additionally, traditional LSTM relies solely on unidirectional temporal information flow, meaning that predictions at the current time step are based only on past information and cannot fully utilize future time steps' data. This limitation may reduce its ability to capture temporal features effectively. BiLSTM further extends the modeling capability of LSTM by introducing a bidirectional information propagation mechanism (forward and backward), allowing it to learn temporal dependencies from both past and future time steps, thus capturing contextual information in time series data more comprehensively. The core idea is to use a forward LSTM to process the time series from the start to the end, while a backward LSTM processes the time series from the end to the start. The forward and backward hidden states are then fused to enhance the model's ability to capture global temporal features. In the cooling load forecasting task in manufacturing plants, the load data often exhibit complex characteristics such as multi-peak and periodic fluctuations. BiLSTM, by merging forward and backward information, improves the model's ability to represent these features, enabling it to adapt well to load changes at different time scales. The BiLSTM structure is shown in Figure 2. In this figure, BiLSTM takes multivariate time series data with a history step size of t and N variables as input, learns the forward hidden state \(\overset{\rightarrow}{h}_{t}\) and the backward hidden state \(\overset{\leftarrow}{h}_{t}\) through the forward and backward LSTMs, and then concatenates and fuses them into \(H_{t}\). The final output \(y_{t}\) is obtained through the output layer. The corresponding formulas are as follows:  
\[\left\{ \begin{array}{l}\overset{\rightarrow}{h}_{t}=LSTM_{forward}(x_{t}, \overset{\rightarrow}{h}_{t-1})\\ \overset{\leftarrow}{h}_{t}=LSTM_{backward}(x_{t},\overset{\leftarrow}{h}_{t- 1})\\ H_{t}=[\overset{\rightarrow}{h}_{t},\overset{\leftarrow}{h}_{t}]\\ y_{t}=Outputlayer(H_{t})\end{array} \right. \tag{2}\]  
Figure 1: Structure of the LSTM Module. BiLSTM is essentially still a type of recurrent network, which processes multivariate time series data on a time-step basis. Each time step contains multiple feature values, and through bidirectional information interaction (forward and backward), it can effectively capture the temporal dependencies in multivariate time series.