# Turbofan Engine Health Prediction Model Based on ESO-BP Neural Network
## 4 Simulation Experiments and Result Analysis
### Ablation Experiment  
An ablation experiment was conducted to assess the impact of each enhancement approach on the BP model. The experiment compared the baseline BP(LM) with SO-BP(LM)  
Figure 12: Scatter of predicted RHI and real RHI values: (**a**) is the CNN model; (**b**) is the RNN model; (**c**) is the LSTM model; and (**d**) is the ESO-BP model. The blue scatter plot represents the comparison between the predicted RHI values and the real RHI values. The \(y\)-axis denotes the predicted RHI values, while the \(x\)-axis presents to the real RHI values of FD001.  
Figure 13: Comparing the RHI degradation prediction results of different models on four engines: (**a**) is the No. 100 engine in the test set of FD001; (**b**) is the No. 124 engine in the test set of FD002; (**c**) is the No. 39 engine in the test set of FD003; and (**d**) is the No. 47 engine in the test set of FD004.  
and ESO-BP(LM). Table 6 presents the average evaluation metrics of 100 trials for BP(LM), SO-BP(LM), and ESO-BP(LM).  
The evaluation metrics derived from 100 trials demonstrate that SO optimized BP neural networks on the FD001, FD002, and FD003 sub-datasets of the C-MAPSS. ESO further enhanced the optimization effect and showed improvement on the FD004 subset, where SO-BP performed poorly. In order to visualize the improvements of SO and ESO on BP neural networks, Figure 14 shows the RMSEs of BP(LM), SO-BP(LM), and ESO-BP(LM).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dataset** & **Model** & **RMSE** & \(\mathbf{R^{2}}\) & _Score_ \\ \hline \multirow{3}{*}{FD001} & BP & \(6.070\times 10^{-2}\) & \(9.304\times 10^{-1}\) & \(4.891\times 10^{-1}\) \\  & SO-BP & \(6.065\times 10^{-2}\) & \(9.305\times 10^{-1}\) & \(4.859\times 10^{-1}\) \\  & ESO-BP & \(\mathbf{6.040\times 10^{-2}}\) & \(\mathbf{9.311\times 10^{-1}}\) & \(\mathbf{4.859\times 10^{-1}}\) \\ \hline \multirow{3}{*}{FD002} & BP & \(6.528\times 10^{-2}\) & \(9.275\times 10^{-1}\) & \(1.380\times 10^{-2}\) \\  & SO-BP & \(6.524\times 10^{-2}\) & \(9.276\times 10^{-1}\) & \(1.380\times 10^{-2}\) \\  & ESO-BP & \(\mathbf{6.515\times 10^{-2}}\) & \(\mathbf{9.278\times 10^{-1}}\) & \(\mathbf{1.374\times 10^{-2}}\) \\ \hline \multirow{3}{*}{FD003} & BP & \(6.779\times 10^{-2}\) & \(9.193\times 10^{-1}\) & \(7.187\times 10^{-1}\) \\  & SO-BP & \(6.761\times 10^{-2}\) & \(9.197\times 10^{-1}\) & \(7.182\times 10^{-1}\) \\  & ESO-BP & \(\mathbf{6.744\times 10^{-2}}\) & \(\mathbf{9.202\times 10^{-1}}\) & \(\mathbf{7.146\times 10^{-1}}\) \\ \hline \multirow{3}{*}{FD004} & BP & \(7.546\times 10^{-2}\) & \(9.010\times 10^{-1}\) & \(1.973\times 10^{-2}\) \\  & SO-BP & \(7.610\times 10^{-2}\) & \(8.993\times 10^{-1}\) & \(1.994\times 10^{-2}\) \\ \cline{1-1}  & ESO-BP & \(\mathbf{7.529\times 10^{-2}}\) & \(\mathbf{9.014\times 10^{-1}}\) & \(\mathbf{1.970\times 10^{-2}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Averages of evaluation metrics for the ablation experiment. The bold number represents the best model.  
Figure 14: Boxplots RMSE for BP, SO-BP, and ESO-BP over 100 trials: (**a**) is the RMSE on FD001; (**b**) is the RMSE on FD002; (**c**) is the RMSE on FD003; and (**d**) is the RMSE on FD004. The box represents the  
interquartile range, which is the middle 50% of the data. The bottom and top edges of the box correspond to the first quartile and third quartile, respectively. The line inside the box represents the median or the second quartile, which is the middle value of the dataset. The height of the box is the interquartile range. The circles represent the outliers. The \(y\)-axis denotes the RMSE values, while the \(x\)-axis presents different models.