## 3 Method
### State Correction  
The range of each variable is strictly limited based on the meteorological variables' physical characteristics; for instance, the humidity should be within \([0,100]\). These ranges are also influenced by the location of the observation station; for example, the temperature at the North Pole should be lower than at the equator. Furthermore, certain variables are interdependent; when precipitation occurs, the humidity is typically high, and high wind velocities often occur together with lower temperatures. These characteristics lead to strict constraints on the ranges of variable values, and result in a high risk of unreasonable states occurring during training and forecasting due to the randomness of model parameter optimizations, such as high precipitation and low humidity at the same time. Therefore, we propose that providing extra monitoring of the predicted states and correcting the unreasonable ones can lead to more reliable prediction results and more robust models. After obtaining the constrained prediction \(Y_{mc}\), we project \(Y_{mc}\) from the temporal domain to the hidden space by a fully connected layer \(f_{sp}\) and a nonlinear activation function \(\sigma\):  
\[h_{s}=\sigma(f_{sp}(Y_{mc})) \tag{10}\]  
where \(h_{s}\in\mathbb{R}^{H}\) is the representation of each state and \(H\) is the predefined dimension of the hidden space. To avoid overfitting and make the model more robust, we introduce a dropout layer after the hidden representation \(h_{s}\), as previously deployed in the MCC module network:  
\[h_{s}=Dropout(h_{s}). \tag{11}\]  
The SC module focuses on ensuring that each predicted state adheres to the physical constraints of the meteorological variables. Unlike the MCC module, which operates based on changes in the variables, the SC module directly examines each predicted state at each time step independently. This direct projection of each time step's state into the hidden space allows the model to assess whether that state is reasonable irrespective of how it was reached. To enhance the MLP's ability to capture the state's subtle features among the large scale of meteorological states,we introduce a differential operation on representation \(h_{s}\) inspired by the differential transformer [44], as demonstrated in Figure 3. This differential operation \(diff\) can be calculated as follows:  
\[diff(h_{s})=\exp(\lambda_{1})\odot h_{s}-\exp(\lambda_{2})\odot h_{s} \tag{12}\]  
where \(\lambda_{1},\lambda_{2}\in\mathbb{R}^{P\times H}\) are learnable matrices and \(\odot\) denotes element-wise multiplication. Similar to a differential amplifier [45], this differential operation acts as a feature enhancement mechanism, helping to distinguish important patterns from noise in the hidden representation. By applying different learnable weights (\(\exp(\lambda_{1})\) and \(\exp(\lambda_{2})\)) to the same hidden representation \(h_{s}\), the operation can selectively amplify subtle but important features while suppressing common noise patterns. Then, we deploy another fully connected layer \(f_{sc}\) to judge the state and generate a correction term \(C_{sc}\):  
\[C_{sc}=f_{sc}(h_{s}). \tag{13}\]Finally, we add the correction term \(C_{sc}\) to the constrained prediction \(Y_{mc}\) in order to adjust the states at each time step and generate the final prediction \(Y\):  
\[Y=Y_{mc}+C_{sc}. \tag{14}\]  
Similar to the MCC module, the SC module learns through the gradient propagation mechanism during training. The gradient flow for the SC module is as follows:  
\[\theta_{SC}\leftarrow\theta_{SC}-\eta\frac{\partial\mathcal{L}}{\partial\theta _{SC}} \tag{15}\]  
where \(\theta_{SC}\) represents the learnable parameters in the SC module, including the differential operation parameters \(\lambda_{1}\) and \(\lambda_{2}\).