# Turbofan Engine Health Prediction Model Based on ESO-BP Neural Network
## 5 Discussion  
The objective of training a neural network is to fine-tune weights and biases to find a parameter set that aligns the predicted values with the actual values as closely as possible [44]. Typically, this process involves finding the global optimal solution, which requires determining the combination of parameters that minimizes the loss function across the entire parameter space. The exceptional prediction performance of the ESO-BP model is attributed to three primary factors: optimizing the initial weights and biases through ESO, employing the RHI metric for turbofan engine degradation, and selecting LM as the training function for BP models.  
1. Initializing neural network weights was demonstrated to accelerate convergence, prevent entrapment in local optima, simplify network architecture, identify feature importance, and improve prediction performance [45]. In this study, adjustments to the initial weights and biases were applied before training to improve the convergence accuracy and stability of the BP neural networks. ESO exhibited a superior convergence compared to SO in benchmark functions. Similarly, in the ablation experiment, ESO-BP outperformed SO-BP. This suggests that providing better initial weights and biases results in higher accuracy and stability in BP neural networks. ESO determined the initial weights and biases that minimized the RMSE after its iteration process, preventing the BP neural network from easily becoming stuck in local optima and experiencing stability issues due to random parameter selection.
2. In contrast to RUL prediction, [PERSON] et al. [46] also developed a health index prediction model for turbofan engines in 2023. According to their study, the average accuracy of 10 trials for hybrid methods was significantly lower than that of single methods. Several methods were combined to create multiple new models, including empirical mode decomposition (EMD), variational mode decomposition (VMD), scale-adaptive attention mechanism (SAA), dynamic step size-based fruit fly optimization algorithm (DSSFOA), bidirectional long short-term memory network (BiLSTM), support vector regression (SVR), and LSTM. The average accuracies of the FD002 sub-dataset are compared in Table 7.
3. The Levenberg-Marquardt algorithm is particularly effective in training BP neural networks for regression problems [47]. The LM algorithm provides numerical solutions for nonlinear minimization. It combines the strengths of the Gauss-Newton algorithm and the gradient descent method by dynamically adjusting parameters during execution to address the limitations of both methods. When the gradient drops rapidly, the LM algorithm behaves more like the Gauss-Newton algorithm. Otherwise, it behaves more like gradient descent [43].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Number of Trial** & **RMSE** \\ \hline EMD-SAA-BiLSTM & 10 & 0.101 \\ VMD-SAA-BiLSTM & 10 & 0.082 \\ VMD-DSSFOA-SVR & 10 & 0.156 \\ VMD-DSSFOA-LSTM & 10 & 0.093 \\ VMD-DSSFOA-SAA-LSTM & 10 & 0.086 \\ ESO-BP & 100 & 0.065 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparison of RMSE for hybrid methods [46] and ESO-BP model.