# Time Series Prediction of Sea Surface Temperature Based on an Adaptive Graph Learning Neural Model
## 2 Materials and Methods
### Adaptive Graph Learning Network (AGLN)  
The graph learning module is designed to mine adaptive adjacency matrices driven by the data. However, in terms of time series prediction, most of the existing graph neural network methods based on mining adjacency matrices rely heavily on predetermined graph structure and cannot update graph structure over time during training, that is, adjacency matrix A needs to be calculated according to a distance function or similarity function before input to the model. Firstly, this method of calculating violence requires a great deal of domain knowledge. Secondly, predefined graph structures containing only explicit spatial information cannot mine hidden spatial dependencies for this prediction task, which may lead to considerable bias in the final prediction. Finally, predefined graph structures are not portable or compatible with other prediction tasks. To solve the above problems, the graph learning module proposed herein is an adaptive graph learning network (AGLN) driven by raw data, which is specifically used to automatically mine the hidden interdependence in  
Figure 1: Diagram of the overall framework of the adaptive graph learning neural model (AGLNM). TC module represents the time convolution module, while GC module is the graph convolution module. Y\({}_{\text{real}}\) represents the real SST value sequence, Y\({}_{\text{pre}}\) represents the predicted SST value sequence, L\({}_{\text{pre}}\) represents the loss function of SST prediction, L\({}_{\text{GL}}\) represents the graph loss function of the graph learning layer, and L\({}_{\text{AGLNM}}\) represents the composite loss function of the model AGLNM. E\({}_{\text{A}}\) is the SST space node embedding, and A represents the adaptive adjacency matrix. real data. The adjacency matrix A containing the graph structure information is calculated by Formula (4). Specific instructions are as follows:  
\[\widetilde{A}=D^{-\frac{1}{2}}AD^{-\frac{1}{2}}=Softmax\left(ReLU\left(E_{A}\cdot E _{A}^{T}\right)\right) \tag{4}\]  
where \(A\) is the adjacency matrix, \(D\) is the degree matrix, \(E_{A}\in R_{N\times d}\) represents the node embedding matrix randomly initialized by AGLN for all nodes which can be learned and updated through the training process, \(N\) represents the number of nodes or spatial points, and \(d\) represents the dimension of node embedding. The transition matrix \(\widetilde{A}\) is the adaptive matrix obtained after the normalization of adjacency matrix \(A\) by the softmax function. It is worth noting that we directly update and calculate the dot product \(\widetilde{A}\) of adjacency matrix \(A\) and Laplace matrix \(L\), instead of generating adjacency matrix \(A\) and Laplace matrix \(L\) separately, which can reduce the computational overhead in the iterative training process. In addition, the AGLN adds the following graph loss mechanism to continuously update and optimize the adjacency matrix \(A\) toward the real spatial dependence of the SST data. \[L_{GL}=\sum_{i=0,j=0}^{N}\|x_{i}-x_{j}\|_{2}^{2}\widetilde{A}_{ij}+\gamma\| \widetilde{A}\|_{F}^{2} \tag{5}\]  
where \(\widetilde{A}_{ij}\in R_{N\times N}\) represents the spatial dependence relation between node \(i\) and node \(j\), \(\|x_{i}-x_{j}\|_{2}^{2}\) is the calculation formula for the dependence relation between space point \(i\) and \(j\). The smaller the value is, the larger the value of transition matrix \(\widetilde{A}_{ij}\) will be. Due to the simple property of transition matrix \(\widetilde{A}\), the second term in the formula can control the sparsity of the learned adjacency matrix \(A\). It can be seen from Formula (5) that \(L_{GL}\) is a graph loss function driven by the spatial node data. Therefore, minimizing \(L_{GL}\) enables the AGLNM to adaptively mine the real spatial correlation hidden in SST data. However, minimizing the value of the graph loss function \(L_{GL}\) alone may only provide a general solution.