## 3 Method
### Multi-Variants Correlation Constraint  
There is a distinct correlation among different meteorological variables (e.g., temperature, precipitation) that play important roles in climate change [41]. Thus, we consider that offering additional insights on these relationships may help the model to make more accurate predictions. However, unlike some other multivariate time series data, such as traffic flow and electricity demand, the correlation patterns among meteorological variables are primarily or directly based on physical properties [42] which are strict, specific, stable, and hard to modify or change across the time. However, some methods focus on whole sequences or patches of different channels. For instance, iTransformer considers every channel's sequence as a token, which may introduce irrelevant temporal variances and lead to less effective capture of these temporally independent physical correlation patterns. To address the above issues, we design the Multi-variants Correlation Constraint (MCC) module, which is deployed after the backbone model on the channel dimension of the forecasting series. The detailed architecture of the MCC's network is shown in Figure 2.  
After the backbone generates the initial prediction \(Y_{i}\), we first define the last observed time step \(X_{O}\) as the reference point and representation of the historical states, then calculate the bias between the initial prediction \(Y_{i}\) and \(X_{O}\):  
\[B=Y_{i}-X_{O} \tag{2}\]  
where \(B\in\mathbb{R}^{P\times N}\) is the bias matrix. The MCC module leverages the bias matrix \(B\) as a physically meaningful representation of how each variable evolves from the reference state. By using the change relative to a reference point (\(X_{O}\)) rather than absolute values, the module is able to effectively capture the temporal dynamics of weather evolution. Figure 2: Architecture of the MCC constraint module deployed in an MLP. Then, we deploy a fully-connected layer \(f_{mp}\) and a nonlinear activation function \(\sigma\) on the channel dimension of \(B\) to project the bias of every time steps into the hidden space:  
\[h_{mc}=\sigma(f_{mp}(B)) \tag{3}\]  
where \(h_{mc}\in\mathbb{R}^{P\times H}\) is the hidden representation of the variable correlation and \(H\) is the predefined dimension of the hidden space. We deploy a dropout layer after the hidden representation \(h_{mc}\) to avoid overfitting [43]:  
\[h_{mc}=Dropout(h_{mc}). \tag{4}\]  
The fully connected layer \(f_{mp}\) learns the complex interdependencies between changes in different meteorological variables through its learnable weights. For example, it can model how changes in temperature relate to corresponding changes in humidity, air pressure, and other variables. The nonlinear activation function \(\sigma\) enables modeling of the nonlinear patterns inherent in complex and dynamic meteorological systems. Then, another fully connected layer \(f_{mc}\) is introduced to generate the constraint term \(C_{mc}\):  
\[C_{mc}=f_{mc}(h_{mc}). \tag{5}\]  
Next, we add the constraint term \(C_{mc}\) to the initial prediction's bias \(B\) to generate the constrained bias \(B_{mc}\):  
\[B_{mc}=B+C_{mc}. \tag{6}\]  
Finally, we convert the constrained bias \(B_{mc}\) back to a time series by adding the reference point \(X_{O}\):  
\[Y_{mc}=B_{mc}+X_{O}. \tag{7}\]  
During the process above, the variable correlation constraint term \(C_{mc}\) is inferred by the MLP through the bias matrix \(B\) separately and independently at every time step, thereby avoiding the irrelevant temporal variances and leading to more precise capture of the variable correlation patterns. Unlike fixed correlation rules, the MCC module can dynamically adjust the relationships between meteorological variables through the learnable weights of the linear layers, thereby enhancing the accuracy and reliability of the model's predictions.