## 2 Chebyshev Graph Convolutional Neural Network Modeling Method
### Multi-Task Collaborative Fault Diagnosis Strategy  
In the multi-task collaborative diagnosis module, the model's generalization ability for variable working conditions data is improved by concurrently executing node classification  
Figure 3: (**a**) GAT node feature update process. (**b**) Calculating the attention coefficient of adjacent nodes. (**c**) Node feature fusion update of multiple attention heads. and node reconstruction tasks. In the node classification task, the node representation produced by the second improved ChebyNet layer is refined through supervised learning, enabling the model to better meet classification requirements. In the node reconstruction task, the objective is to map the node features processed by the network back to the original input space, reconstructing node features that closely resemble the original input. This process does not depend on externally labeled data; instead, it utilizes the structural information inherent to the data as a supervisory signal. The output of the node classification task serves as the input for the node reconstruction task, and its accuracy directly influences the quality of the reconstruction task. The node reconstruction task assesses the effectiveness of reconstruction by calculating the reconstruction loss, and this evaluation subsequently aids the node classification task in conducting self-inspection and optimization. When the reconstruction loss is low, it indicates that the model retains sufficient effective information during the feature extraction and transformation process, enhancing the accuracy of the node classification task in identifying node categories; conversely, a larger reconstruction loss may indicate that the model has deviations in feature processing, prompting adjustments and optimizations in subsequent learning. In the forward propagation, for each sample, the network output includes the probability that the sample belongs to the \(i\)-th category and the node reconstruction feature vector (the reconstructed output and the original input features are transformed by the fully connected layer to ensure dimensional consistency), which correspond to the node classification Cross-Entropy Loss and the node reconstruction MSE Loss, respectively. Moreover, the node classification Cross-Entropy Loss measures the difference between the model's predictions for each category and the true labels. The calculation of the classification Cross-Entropy Loss is shown in Equation (14). \[loss=-\frac{n}{i=1}t_{i}\log P_{i} \tag{14}\]  
\[reconstruction\_loss=\frac{1}{m}\sum_{i=1}^{m}\left(reconstructed\_x_{i}-x_{i} \right)^{2} \tag{15}\]  
Here, \(loss\) is the node classification Cross-Entropy Loss; \(t_{i}\) is the value of the \(i\)-th element in the true label vector; \(n\) is the number of categories; and \(P_{i}\) represents the probability vector that the sample is the \(i\)-th category. The node reconstruction loss is calculated using the root MSE, which measures the difference between the output of the model's node reconstruction and the original input, as shown in Equation (15). Here, \(reconstruction\_loss\) is the node reconstruction MSE Loss; \(reconstructed\_x_{i}\) is the \(i\)-th feature dimension of the node reconstruction layer output; \(x_{i}\) is the \(i\)-th feature dimension of the input data; and \(m\) is the number of feature dimensions (experimentally set to 1024). In backpropagation, the previous gradient is first cleared, and the total loss is computed as the sum of the node classification Cross-Entropy Loss and the node reconstruction MSE Loss. Finally, the backpropagation algorithm is employed to compute the gradient with respect to the model parameters, as shown in Equations (16) and (17). Utilizing stochastic gradient descent (SGD) as the optimization method, the gradient is computed, and the model parameters are updated accordingly in the opposite direction. For each parameter \(\theta_{i}^{(t)}\), the update rule is shown in Equation (18).