# An Improved Monte Carlo Reliability Analysis Method Based on BP Neural Network
## 3 Calculation Theory of Improved BP-MCS Method
### The Steps of Reliability Calculation for Improved BP-MCS Method  
Because the BP model constructs the implicit structure-performance function, it takes a longer time to predict the structure response than the explicit structure performance function. When the failure probability is small, the number of MCS calculations reaches more than \(10^{6}\), which makes the calculation time increase dramatically. Considering that the failure probability of the actual engineering structure is very low, meaning the weight of its failure point is very small, then there must be a weight critical value \(\omega_{0}\). When the weight of sample points extracted by MCS is \(\omega_{i}>\omega_{0}\), these sample points are considered reliable, and there is no need to call the BP model to calculate the structural response value at this time. However, when the weight of sample points is \(\omega_{i}<\omega_{0}\), the area contains all failure points, so it is necessary to call the BP model to determine whether failure occurs. Therefore, most of the sample points located in the reliable domain are filtered out in the probability of occurrence, which significantly reduces the number of calls of the BP model and shortens the calculation time.  
The detailed implementation steps of the improved BP-MCS reliability analysis method proposed in this paper are as follows:  
(1) The BP model with high prediction accuracy for the sample points near the failure surface is obtained by training. See 3.3 for details.  
(2) Selecting the weight-critical value \(\omega_{0}\). For the weight-critical value \(\omega_{0}\), it can be selected through the sample points of the training set and the validation set after each BP model accuracy test: Arrange the sample points of the training set and the validation set according to their descending weights, finding the first failure point with \(G<0\), and return the weight \(\omega_{1}\) of the previous sample point with \(G>0\). If there is a sample point very close to \(0^{+}\) before the first failure point, the weight \(\omega_{1}\) of this sample point is returned. To avoid not taking all failure points into account, the amplification factor \(k\) can be added. According to the follow-up research, \(k\) can be 1.1-1.3, then the critical weight value is:  
\[\omega_{0}=k\omega_{1} \tag{11}\]  
(3) Using the improved MCS method to calculate the failure probability. Firstly, \(N_{MC}\) sample points are extracted by MCS, and the weight of each sample point is calculated. When the weight of the sample point is \(\omega_{i}>\omega_{0}\), the sample point is reliable, and there is no need to call the BP model. When \(\omega_{i}<\omega_{0}\), the BP model should be called to judge whether it is failed. If the predicted value \(G^{\prime}(X_{i})<0\), the sample point will be failed. If the total number of failure points is \(N_{f}\), the structural failure probability is:  
\[P_{f}=\frac{N_{f}}{N_{MC}} \tag{12}\](4) Testing the variation coefficient of failure probability according to Equation (2) to ensure that the number of sample points meets the requirements. If \(COV<0.05\), the failure probability is output. Otherwise, go back to step (3) to generate more MCS sample points for calculation, and the initial sample number can be \(10^{6}\). The above iterative process is repeated until the convergence requirement is met, and the final failure probability \(P_{f}\) is output.  
Combined with the training steps of the high-precision BP model, the specific calculation flow of the improved BP-MCS method is shown in Figure 4.