# An Improved Monte Carlo Reliability Analysis Method Based on BP Neural Network
## 4 Examples Verification
### An Explicit Polynomial with a Low Probability of Failure  
Example 2 is an explicit polynomial with a low probability of failure. The random variables \(x_{1}\) and \(x_{2}\) are independent of each other and obey the standard normal distribution. The structural performance function is:  
\[G(x_{1},x_{2})=0.5(x_{1}-2)^{2}-1.5(x_{2}-5)^{3}-3 \tag{14}\]  
For this two-dimensional problem, [PERSON] et al. (2017) used MCS, Importance sampling (IS) method, and AK-IS method to calculate it, [PERSON] et al. (2018) used the classical AK-MCS to analyze its reliability, and [PERSON] et al. (2019) directly solved the failure probability of this problem by combining the BP neural network. This paper adopts the improved BP-MCS method to solve the problem, in which the total number of MCS sample points is \(5\times 10^{7}\), and the calculation results are shown in Table 8.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(k\) & \(P_{f}\) & \(\epsilon\) & \(N_{call}\) & **t (s)** \\ \hline
1.0 & \(2.197\times 10^{-3}\) & 1.61\% & 14,438 & 126.7 \\
1.1 & \(2.243\times 10^{-3}\) & 0.45\% & 15,714 & 144.5 \\
1.2 & \(2.239\times 10^{-3}\) & 0.27\% & 17,502 & 151.6 \\
1.3 & \(2.245\times 10^{-3}\) & 0.54\% & 18,595 & 155.6 \\
1.4 & \(2.248\times 10^{-3}\) & 0.67\% & 20,059 & 164.1 \\
1.5 & \(2.227\times 10^{-3}\) & 0.27\% & 21,163 & 173.3 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Calculation results of failure probability under different amplification coefficients.  
Figure 5: The Improved BP-MCS sample set from example 1.  
It can be seen from Table 8 that the number of sample point calculations of Improved BP-MCS is 77, which is obviously lower than that of MCS, IS, and BP-MCS, but there is still a certain gap with AK-IS and AK-MCS. In terms of surrogate model calls, the algorithm reduces the calculation times of \(5\times 10^{7}\) to 24,155 times, which indicates that the calculation workload reduced by this method is more obvious when the failure probability is small. At the same time, the relative error of failure probability calculated by this method is 0.35%, indicating that the accuracy is high.  
Compared with the BP-MCS algorithm adopted by [PERSON] et al. [24], the Improved BP-MCS algorithm proposed in this paper has a large degree of improvement in accuracy and calculation cost. The specific explicit polynomial and the calculation time of the failure probability of Improved BP-MCS are shown in Table 9. According to Table 9, the calculation time of the Improved BP-MCS is very close to the explicit polynomial, while the calculation time of the Direct BP-MCS may take 2,763,403 s according to the conversion of the calculation time of the Improved BP-MCS. This shows that compared with Direct BP-MCS, the time cost of the Improved BP-MCS is reduced by thousands of times, which proves the high efficiency of the method when the failure probability is low.  
Figure 6 shows the final sample set used in Improved BP-MCS. It can be found that the key points adopted are distributed uniformly globally, and the sample points added by iteration mainly focus on the failure surface. Due to the low failure probability, the weight critical value adopted by the Improved BP-MCS is lower, that is, \(\omega_{0}=7.7\times 10^{-5}\), and the critical region is shown as a circle in Figure 6. A large number of sample points extracted by MCS will be concentrated in the circular region, at which time the sample points are reliable and there is no need to call the BP model for calculation, so the efficiency of this method is explained in space when the failure probability is low.