## 2 Methods
### Sequence-to-Sequence Learning for Temporal Dynamics Modeling  
The gated recurrent unit (GRU) network is a classic type of RNN that is particularly effective at modeling sequential data with complex temporal dependencies. By adaptively updating its hidden state through a gating mechanism, the GRU can selectively remember and forget certain information over time, making it well-suited for time series prediction. The architecture of a typical full gated unit can be expressed as follows:  
\[z_{t}=\sigma_{g}(W_{z}x_{t}+U_{z}h_{t-1}+b_{z}) \tag{6}\]  
\[r_{t}=\sigma_{g}(W_{r}x_{t}+U_{r}h_{t-1}+b_{r}) \tag{7}\]  
\[\hat{h}_{t}=\phi_{h}(W_{h}x_{t}+U_{h}(r\odot h_{t-1})+b_{h}) \tag{8}\]  
\[h_{t}=z_{t}\odot h_{t-1}+(1-z_{t})\odot\hat{h}_{t} \tag{9}\]  
where \(x_{t}\) and \(h_{t}\) are the input vector and output vector at time \(t\), respectively; \(r_{t}\) and \(z_{t}\) refer to the state of reset gate and update gate at time \(t\), respectively; \(\hat{h}_{t}\) represents reset hidden stats at time \(t\) while \(h_{t-1}\) denotes initial hidden states at time \(t-1\); \(W\) and \(U\) are parameter matrices; \(b\) the bias; operator \(\odot\) denotes the Hadamard product; \(\sigma_{g}\) is a sigmoid activation function; and \(\phi_{h}\) is a hyperbolic tangent function. By replacing the matrix multiplication in GRU with the diffusion convolution \(\star_{G}\) over Graph \(G\) defined in Equation (4), the Diffusion Convolutional Gated Recurrent Unit (DCGRU) can be thus expressed as follows:  
\[z^{t}=\sigma_{g}\Bigl{(}\Theta_{z}\star_{G}\Bigl{[}X^{t},L^{t-1}\Bigr{]}+b_{z} \Bigr{)} \tag{10}\]  
\[r^{t}=\sigma_{g}\Bigl{(}\Theta_{z}\star_{G}\Bigl{[}X^{t},L^{t-1}\Bigr{]}+b_{r} \Bigr{)} \tag{11}\]\[\hat{h}^{t}=\phi_{h}\Big{(}\Theta_{h}\star_{G}\left[X^{t},\left(r^{t}\odot L^{t-1} \right)\right]+b_{h}\Big{)} \tag{12}\]  
\[L^{t}=z^{t}\odot L^{t-1}+\left(1-z^{t}\right)\odot\hat{h}^{t} \tag{13}\]  
where \(X^{t}\) and \(L^{t-1}\) are the input and output at time \(t\), respectively. \(\Theta_{z}\), \(\Theta_{r}\), and \(\Theta_{h}\) are the parameters for the corresponding filters. Like GRU, the DCGRU can be utilized to construct the recurrent neural network structures and optimized via the backpropagation through time method, capturing complex temporal dynamics in a variety of applications. Sequence to Sequence (Seq2 Seq) [38] model is a type of neural network architecture that is commonly used for sequence prediction or generation. The fundamental idea of the Seq2 Seq models is to use an encoder-decoder architecture. In the model, the encoder network maps an input sequence into a fixed-length vector representation and the decoder network generates the output sequence based on the vector representation. The DCRNN adopts the Seq2 Seq model architecture with both encoder and decoder implemented by the DCGRU, which enables the network to effectively perform multi-step prediction tasks with temporal dependencies. During the training phase, the encoder of the Seq2 Seq architecture takes the historical time series as input and generates final states, which are then used to initialize the decoder. The decoder is responsible for generating predictions based on the ground truth values that are previously observed. During the testing phase, the model uses its own predictions as inputs instead of the ground truth values. However, the discrepancy between the input distributions during training and testing can decline the model performance. In order to mitigate the potential degradation of performance caused by the discrepancy between input distributions, the DCRNN integrates Scheduled Sampling [35] into the model. Specifically, the DCRNN generates predictions that simultaneously capture the temporal dependencies of the multi-source inputs and the spatial dependencies of the topological features. When training, the model puts ground truth inputs into the decoder. During testing, the model feeds the predictions generated by the decoder as inputs to the next time step.