# PEMFC RUL Prediction for Non-Stationary Time Series Based on Crossformer Model
## 4 Results and Discussion
### Results  
#### 4.2.1 Network Model Comparative Experiment  
Table 1 presents the comparative experimental results of the experimental vehicles at different time step lengths. In this study, the experimental subject was a fuel cell van with the VIN LDPGDBAD3 KF*******. The bolded results indicate the best performance. The input time series length for the experiment was 48, and the Adam optimizer was employed to train and optimize the model, ensuring effective convergence and accurate trend prediction when processing time series data. Observations from the table reveal that the RUL prediction model designed in this study outperforms the other five models in terms of both evaluation metrics, MSE and DTW, in the task of PEMFC RUL prediction. The proposed model, based on the Crossformer architecture, effectively addresses the issue of vanishing or exploding gradients that may occur in LSTM-based models. Additionally, it preserves the non-stationary information in the original time series while considering the dependencies between multiple features in multivariate time series. According to the analysis of the MSE metric, the model proposed in this paper significantly outperforms other mainstream models across prediction time steps ranging from 8 to 64 when the input sequence time step is set at 48. Specifically, compared to the Crossformer model, there is an average performance improvement of 17.73% across all prediction time steps; within the range of 8 to 24 time steps, the average improvement is 15.69%; and within the range of 32 to 64 time steps, the performance improves by an average of 19.77%. These results indicate that the model proposed in this study exhibits superior performance in the MSE, particularly over longer prediction time steps, in tasks related to PEMFC RUL prediction. In this study, TILDE-Q was introduced as the loss function, which is more effective in handling unstable data, especially for the dataset used in this research, which was collected from real-world vehicles in operation. At the same time, dynamic time warping (DTW) was employed to measure the similarity between two distributions. According to the analysis of the DTW metric, as the time step length increases, the DTW values also show a significant upward trend in the PEMFC RUL prediction models used in the experiment. Notably, across different prediction time steps, the model proposed in this study continues to perform exceptionally well, outperforming other mainstream models. Compared to the Crossformer model, the performance of the proposed model improves by an average of 8.98% across prediction time steps ranging from 8 to 64. This indicates that the model proposed in this paper demonstrates superior performance in capturing the similarity between the predicted sequence and the actual sequence in PEMFC RUL prediction tasks. In the combined MSE and DTW evaluation metrics, it is evident that the model proposed in this paper not only significantly enhances the accuracy of the predictions in PEMFC RUL prediction tasks but also improves the conformity of the predicted values to the actual shapes. This holds significant importance and value for both researchers and industry practitioners.