# Turbofan Engine Health Prediction Model Based on ESO-BP Neural Network
## 3 Model Construction
### BP Neural Network  
The BP neural network model's performance is significantly impacted by the number of hidden layers, number of neurons in the hidden layer, initial weights, and biases. The weights indicate the strength of the connections between neurons, while the biases are additional learnable parameters for neurons that introduce shifts and offsets to the neuron inputs, adjusting the input to the activation function [14].  
This study consistently used a single hidden layer for the BP neural networks to control the variable. The number of neurons in the hidden layer was determined using the empirical formula, as shown in Equation (10) [15].  
\[h_{n}=\sqrt{i_{n}+o_{n}}+\alpha \tag{10}\]  
In the equation, \(h_{n}\) represents the number of neurons in the neural network's hidden layer, \(i_{n}\) represents the number of neurons in the input layer, \(o_{n}\) represents the number of neurons in the output layer, and \(\alpha\) is a positive integer with a value ranging from 1 to 10.  
Figure 5: Normalized features of the No. 100 engine in the training set of FD001. (**a**) Shows the ten features with an increasing trend and (**b**) shows the five features with a decreasing trend.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Feature** & **Pearson Correlation Coefficient** & **Feature** & **Pearson Correlation Coefficient** \\ \hline CYCLE & \(1\) & EPR & \(\underline{0}\) \\ H & \(-0.037\) & PS30 & \(0.840\) \\ MA & \(0.013\) & PHI & \(-0.824\) \\ TRA & \(\underline{0}\) & NRF & \(0.789\) \\ T2 & \(\underline{0}\) & NRC & \(-0.057\) \\ T24 & \(0.692\) & BPR & \(0.739\) \\ T30 & \(0.672\) & FARB & \(\underline{0}\) \\ T50 & \(0.809\) & HT\_BLEED & \(0.682\) \\ P2 & \(\underline{0}\) & NF\_DMD & \(\underline{0}\) \\ P15 & \(0.121\) & PCNFR\_DMD & \(\underline{0}\) \\ P30 & \(-0.787\) & W31 & \(-0.743\) \\ NF & \(0.780\) & W32 & \(-0.714\) \\ NC & \(0.459\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Pearson correlation coefficients between number of operating cycles and all the 25 features of the NO. 100 engine in the training set of FD001. The underlined values present the redundant features.  
The optimal value of \(\alpha\) for finding the number of neurons in the hidden layer was found using grid search. The initial weights and biases of the BP neural network were optimized using ESO. The optimization problem for the weights and biases was formulated in a high-dimensional space, and the fitness function was based on the root mean square error of the BP neural network on training sets [13]. Figure 6 illustrates the optimization process. Equations (11) and (12) are used to calculate the dimensions and fitness values [14].  
\[d=(i_{n}+1)*h_{n}+(h_{n}+1)*o_{n} \tag{11}\]  
\[f_{i}=\frac{\sum_{i=1}^{M}(t_{i}-\hat{t}_{i})^{2}}{M} \tag{12}\]  
Here, \(d\) signifies the search space dimension of ESO, \(f_{i}\) is the fitness value of the \(i\)-th iteration of ESO, \(M\) indicates the number of samples in the training set, \(t_{i}\) denotes the actual value of the \(i\)-th sample, and \(\hat{t}_{i}\) represents the model's predicted value for the \(i\)-th sample.