# An Improved Monte Carlo Reliability Analysis Method Based on BP Neural Network
## 4 Examples Verification
### Reliability Problem of Series System  
Example 1 considers a widely used benchmark problem with weak nonlinearity, which is a series system with four branches. The structural performance function is defined as:  
\[G(x_{1},x_{2})=min\left\{ \begin{matrix}g_{1}=3+0.1{(x_{1}-x_{2})}^{2}-{(x_{1}+ x_{2})}/{\sqrt{2}}\\ g_{2}=3+0.1{(x_{1}-x_{2})}^{2}+{(x_{1}+x_{2})}/{\sqrt{2}}\\ g_{3}=x_{1}-x_{2}+7/\sqrt{2}\\ g_{4}=x_{2}-x_{1}+7/\sqrt{2}\end{matrix} \right\} \tag{13}\]  
where \(x_{1}\) and \(x_{2}\) are two independent random variables, both of which obey the standard normal distribution. Based on its weak nonlinearity, many scholars had adopted different improved methods to analyze this explicit problem. The calculation results of the proposed method and  
Figure 4: The calculation flow of the Improved BP-MC algorithm. related literature results are shown in Table 5. As can be seen from Table 5, some commonly used methods are mainly acceleration methods based on the Kriging model and the deep learning theory (DRL). These methods focus on how to reduce \(N_{FEM}\), but do not take into account the computational burden brought by the calculation of a large number of MCS sample points. The sampling times of MCS adopted by the proposed method in this paper is \(10^{6}\), and the BP model adopted by Direct BP-MCS in the table is consistent with the Improved BP-MCS. According to Table 5, AK-MCS-IRS greatly reduces the calculation times of mathematical formulas compared with other methods, but this method is only applicable to low-dimensional problems [28]. The Improved BP-MCS algorithm proposed in this paper is only 88 times, which is lower than some commonly used AK-MCS improvement algorithms such as AK-MCS-U and AK-MCS-EFF. The DRL method based on deep learning theory, which is as high as 2597 times, is relatively poor in practicality. By comparing the call times of the surrogate model, it is found that the Improved BP-MCS algorithm is reduced from \(10^{6}\) to 15,714, while other methods are consistent with MCS, so the computation workload can be greatly reduced. At the same time, the relative error of the failure probability obtained by the Improved BP-MCS is only 0.45%, indicating that the calculation accuracy is higher. Based on the MATLAB platform, the failure probability calculation time of example 1 explicit polynomial, Direct BP-MCS, and Improved BP-MCS are calculated, and the calculation results are shown in Table 6. By comparing the calculation time of explicit polynomial and Direct BP-MCS, it can be seen that the time cost of calculating the failure probability by using the implicit structural performance function is increased by about 373 times compared with the explicit structural performance function. However, when the improved method is adopted, the calculation time is only increased by about seven times. Figure 5 shows the sample set finally used by Improved BP-MCS. It can be found that the key points used are evenly distributed globally, and the sample points added by iteration are mainly concentrated near the failure surface. For the amplification factor \(k\), the failure probability is calculated for \(k\) from 1.0 to 1.5, and the calculation results are shown in Table 7. As can be seen from Table 7, with the increase of the amplification factor, the number of calls of the BP model also gradually increases, resulting in the increase of the calculation time of the failure probability, but its relative error gradually becomes \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & \(\mathbf{N_{call}}\) & \(\mathbf{t(s)}\) \\ \hline Polynomial & \(10^{6}\) & 19.6 \\ Direct BP-MCS & \(10^{6}\) & 7325.1 \\ Improved BP-MCS & 15,714 & 144.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Example 1 calculation time of failure probability by different methods.