## 3 Methodology
### Encoder-Decoder Forecasting  
The encoder-only forecasting architecture is easier to implement with a lower computation complexity. This schema focuses on capturing the potential links between variables at the time steps before and after. The projection layer simply takes the basic MLP mapping the higher-order representation to the desired output dimension. Considering that MLPs process each time step independently, lacking mechanisms to model temporal relationships between predicted steps, it is problematic for autoregressive decoding, where each prediction depends on prior outputs. On the other side, MLPs cannot maintain a hidden state across predictions, making it difficult to enforce coherence in multi-step forecasts (e.g., trend continuity or seasonality alignment). To deal with this issue, the decoder layer is introduced to describe the sequence relation of time-series data better based on the idea of auto-encoder.  
Similarly, the encoder layer contains two components of the embedding and TrmBlock layer, which is consistent with the encoder-only framework. The embedding layer converts the observed time-series data input into a number of tokens for each variable characteristic in Equation (6). Then, the variate token is fed into the transformer components to learn the high-order representation as shown in Equation (7).  
\[H^{0}:,n=Embedding(X:,n) \tag{6}\]  
\[H^{l}=Transformer\left(H^{l-1}\right) \tag{7}\]  
Afterward, we utilize the Long Short-Term Memory (LSTM) as the decoder layer to achieve the complex relationship between the elements and the time evolution law as shown in Equations (8)-(10), where \(I_{t},F_{t},Q_{t}\) denote the input, forget, and output gates. These equations show how the current input \(X_{t}\) and the previous hidden state \(H_{t-1}\) are transformed with weight matrices \(W\) and bias vectors \(b\). Afterwards, the sigmoid activation function \(\sigma\) is utilized for mapping the input to an interval of \((0,1)\). LSTM explicitly models sequential dependencies between predictions, using their recurrent structure to iteratively update the hidden state. Meanwhile, LSTM is more stable than ordinary RNN because its gating mechanism can effectively capture long-term dependence and alleviate the gradient vanishing and explosion problem.  
\[I_{t}=\sigma(X_{t}W_{xi}+H_{t-1}W_{hi}+b_{i}) \tag{8}\]  
\[F_{t}=\sigma\left(X_{t}W_{xf}+H_{t-1}W_{hf}+b_{f}\right) \tag{9}\]  
\[O_{t}=\sigma(X_{t}W_{xo}+H_{t-1}W_{ho}+b_{o}) \tag{10}\]  
In addition, the candidate memory denoted by \(C_{t}\) is calculated by Equations (11) and (12), where \(\widetilde{C_{t}}\) denotes the candidate cell state, and \(tanh\) denotes the hyperbolic tangent activation function. The operator symbol \(\odot\) represents element-wise multiplication (Hadamard product). In addition, \(W_{xc}\) and \(W_{hc}\) denote the weight matrix of the candidate cell state, which is used to linearly transform the concatenation vector of the hidden state at the previous time and the input at the current time, and \(b_{c}\) represents the bias term of the candidate cell state.  
\[\widetilde{C_{t}}=tanh(X_{t}W_{xc}+H_{t-1}W_{hc}+b_{c}) \tag{11}\]  
\[C_{t}=F_{t}\odot C_{t-1}+I_{t}\odot\widetilde{C_{t}} \tag{12}\]  
The forget gate is used to control the memory state of the previous step, and the input gate controls the candidate memory state, as shown in Equation (13).  
\[H_{t}=O_{t}\odot tanh(C_{t}) \tag{13}\]  
Also, the MLP is used to align the decoder results to the desired output \(Y2_{Long}\in\mathbb{R}^{N\times L}\), as shown in Equation (15).  
\[Y2_{Long}=MLP_{Long}\left(\text{H}_{t}^{l+1}\right) \tag{14}\]Afterward, the model is trained independently by minimizing the mean squared error (MSE), which calculates the average squared difference between the predicted values and the target values as shown in Equation (15).  
\[Loss_{2}=\frac{1}{N_{L}}{\sum_{i=1}^{N_{L}}}\big{(}Y_{i}-Y2_{Long_{i}}\big{)}^{2} \tag{15}\]