## 3 Multi-Step Prediction Framework for Multivariate Corrosion Information Based on Adaptive MTGNN  
### Brief Description of the Corrosion Rate Prediction Problem  
The spatial inhomogeneity of the internal environment of the pipeline and the corrosion propagation mechanism lead to the spatial dependence of the corrosion rate at different locations of the pipe wall. The MTGNN-based corrosion rate prediction model proposed in this paper utilizes the graph learning module of MTGNN to adaptively find the latent spatial correlation between the corrosion states of each node to improve the prediction accuracy while capturing the time-dimensional information. Let \(Z_{t}=[z_{1,t},z_{2,t},\cdots,z_{N,t}]\in R^{N\times D}\) denote the historical feature values of the \(N\) corrosion monitoring nodes in the pipeline at time \(t\), where \(z_{i,t}=[z_{i,t},1,z_{i,t  
as the corrosion rate (primary feature), and the rest of the columns are used as auxiliary features. Therefore, the corrosion information dataset of \(T\) time steps for \(N\) monitoring nodes can be represented as \(X=[Z_{1},Z_{2},\cdots,Z_{T}]\in R^{T\times N\times D}\). In order to realize the embedding of corrosion information in the graph, the spatio-temporal information graph of corrosion rate, constructed by treating the corrosion sensor at each location as a corrosion rate monitoring node, is shown in Figure 5. The graph of the corrosion rate is represented as \(G=(V,E)\), where \(V\) is the set of corrosion rate nodes, and \(E\) is the set of edges between corrosion rate nodes. Given historical observations of the multivariate variable \(X=[Z_{1},Z_{2},\cdots,Z_{T}]\), the goal of the model is to predict the sequence of future values \(Y=\left[Z_{T+1},Z_{T+2},\cdots,Z_{T+Q}\right]\). In this paper, we set \(T_{in}=15,T_{out}=6\), i.e., the historical data of the last 15 time nodes are used to predict the corrosion rate of the next 6 time nodes. In order to avoid the limitations of manually defining the graph structure and highlight the adaptability of the dynamic graph structure to the spatial heterogeneity of pipeline corrosion, this paper adopts the adaptive MTGNN as the core framework, and the specific steps of the pipeline corrosion spatio-temporal correlation model constructed in this paper are as follows:  
1. Generate initial node embedding vectors based on the physical location coordinates of the monitoring points and multi-feature data as input to the graph learning layer. 2. Calculate the similarity between nodes through a two-branch structure and use the graph learning layer to generate a dynamic adjacency matrix that captures the implicit spatial dependencies between nodes. 3. A top-k strategy is used to select the critical neighbors of each node, and non-critical connections are set to zero to optimize the matrix sparsity and reduce the computational complexity. 4. Input the dynamic adjacency matrix into the graph convolution module and integrate the multi-feature information of the node itself and its neighbors through the mix-hop propagation layer to generate the spatial feature representation. 5. Alternately stack the graph convolution module and the temporal convolution module to extract spatial and temporal features, respectively, retain the original information through residual connections, skip connections to transfer features across layers,  
Figure 5: Spatio-temporal information map of corrosion rate. and finally output the predicted corrosion rate that integrates spatial and temporal information. To address the problem of error accumulation in multi-step prediction, this paper proposes a \"chunked progressive\" training strategy based on the idea of \"curriculum learning\" [30]. The method gradually expands the prediction horizon of the model by decomposing the multi-step prediction task into multiple sub-blocks.