## 5 Discussion  
The results presented in Section 4 demonstrate the efficiency and accuracy of the proposed PSO-BPNN-based reliability analysis approach. To further justify these findings and validate the effectiveness of the proposed method, a detailed discussion is provided below:  
(1) Comparison with existing methods.  
Compared to traditional reliability analysis methods, the proposed PSO-BPNN approach demonstrates significant advantages. Traditional methods often face challenges when dealing with complex nonlinear implicit functions, primarily due to their limitations in capturing global optimization solutions and their susceptibility to overfitting. In contrast, the PSO-BPNN method utilizes the global optimization capability of the PSO algorithm to optimize the weights and biases of the BPNN. This enhances the neural network's ability to approximate complex implicit limit state functions while effectively avoiding local extrema. Furthermore, the integration of the MCS method ensures robust and accurate estimation of failure probabilities.  
(2) Efficiency and computational cost.  
Numerical results reveal that the proposed method achieves a high level of computational efficiency. For example, in the face of high nonlinearity or high-dimensional situations, the computational time was significantly reduced compared to conventional methods, all while maintaining accuracy. This improvement in computational efficiency can be attributed to the optimized neural network training process.  
(3) Improvements in accuracy.  
The accuracy of the proposed approach has been validated through numerical examples, with results closely aligning with benchmark solutions obtained using traditional methods. The integration of PSO ensured that the neural network effectively avoided overfitting, a common challenge in reliability analysis when employing machine learning models.  
(4) Advantages in practical applications.  
The practical utility of the proposed method was demonstrated in the reliability analysis of an aero-engine turbine structure, a complex engineering problem characterized  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Failure Probability P\({}_{f}\)** & **Number of Samples \(K\)** & \(\mathbf{R}^{2}\) \\ \hline Proposed & \(2.43\times 10^{-3}\) & 1000 & 99.993\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Failure probability calculated by PSO-BPNN method.  
by pronounced nonlinearities. The results confirm that the method effectively addresses the challenges posed by such real-world applications, providing accurate and reliable outcomes. This highlights its potential as a valuable tool for engineering reliability assessments, especially in aerospace engineering, where implicit nonlinear functions are prevalent.  
Limitations and future directions.  
Although the proposed approach performed well in the presented examples, further research is required to evaluate its scalability for even more complex systems. Additionally, the integration of adaptive sampling techniques into the PSO-BPNN framework could further improve its computational efficiency. Future work could also focus on comparing the proposed method with other advanced machine learning-based approaches to better understand its relative performance and potential advantages.  
In conclusion, the proposed PSO-BPNN-based reliability analysis approach features both accuracy and efficiency, while offering substantial advantages in addressing complex nonlinear implicit reliability problems in engineering applications.