## 3 Results and Discussion
### Performance Decay Prediction of the DEA Mode
#### 3.1.1 Comparison of the ANN and M-ANN  
In the DEA mode, the PEM fuel cell performance firstly increases due to the self-humidification effect, and then gradually decreases due to the nitrogen crossover and accumulation in the anode, as shown for the data in the 12 processes. More physical explanation could be found in our previous work [21]. Comparison of one-hidden-layer ANN and one-hidden-layer M-ANN predictive values and true values of current density change evolution of the PEM fuel cell in the DEA mode are shown in Figures 2 and 3, respectively. Comparison of two-hidden-layer ANN and two-hidden-layer M-ANN predictive values and true values of current density change evolution of the PEM fuel cell in the DEA mode are shown in Figures 4 and 5. (a)-(h), (i,j) and (k,l) represent the train set, validation set, and test set, respectively. The trend of current density firstly increasing and then decreasing is predicted by the ANN-based model for all the 12 processes. It is worth mentioning that it takes around 4 hours' calculation for the small workstation to obtain the best one-hidden-layer ANN model through grid search, while it takes around 70 h to obtain the best two-hidden-layer ANN model. Prediction performance, including the mean and maximum relative errors of different modeling methods for the DEA mode is listed in Table 4.  
Figure 2: One-hidden-layer ANN with ReLU activation function. Comparison of predictive values and true values of current density change evolution for the PEM fuel cell with the DEA mode. (**a**–**I**) represent the 12 transient processes. (**a**–**h**), (**i**,j) and (**k**,l) are the train set, validation set, and test set, respectively. Figure 4: Two-hidden-layer ANN with ReLU activation function. Comparison of predictive values and true values of current density change evolution for the PEM fuel cell with the DEA mode. (**a**–**I**) represent the 12 transient processes. (**a**–**h**), (**i**,**j**) and (**k**,**l**) are the train set, validation set, and test set, respectively. Figure 3: One-hidden-layer M-ANN with ReLU activation function. Comparison of predictive values and true values of current density change evolution for the PEM fuel cell with the DEA mode. (**a**–**I**) represent the 12 transient processes. (**a**–**h**), (**i**,**j**) and (**k**,**l**) are the train set, validation set, and test set, respectively. The mean and maximum relative errors by one-hidden-layer ANN are 0.725% and 1.563% that are much larger than that by one-hidden-layer M-ANN of 0.198% and 0.803%. The mean and maximum relative errors by two-hidden-layer ANN are 0.341% and 0.932% that are much larger than that by two-hidden-layer M-ANN of 0.158% and 0.534%. Although the prediction degradation process of M-ANN includes the error of the MPR model to predict the initial current density and the error of the ANN model to predict the current density change. However, from the results of M-ANN, this is tolerable. It can be seen that the prediction curves by ANN show more obvious error compared with the true curves than those by M-ANN. The reason why M-ANN can obtain better prediction results than ANN may be as follows:  
1. The MPR method solves the problem of initial current density prediction under various working conditions. This is because each operating condition has a different starting point for its current density due to different initial conditions, which is the reason for the poor effectiveness of ANN in predicting the overall degraded current density. ANN can also be used to predict initial current density at various operating conditions. the initial current density dataset has only 12 samples. It is well known that the ANN model needs a large amount of data to be trained to get better results.