# An Improved Monte Carlo Reliability Analysis Method Based on BP Neural Network
## 3 Calculation Theory of Improved BP-MCS Method
### High-Precision BP Neural Network Prediction Model  
On the limit state surface, the structure performance function value is 0. In the reliable region, the structure performance function value is positive. In the failure region, the structure performance function value is negative. When using the BP model to calculate MCS reliability, we only need to pay attention to whether the response value of the sample points is reliable, but not to its value. Therefore, it is only necessary to ensure and improve the prediction accuracy of the BP model near the limit state surface on the basis of uniform coverage of the global region, then the BP model can be used to judge whether the sample  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Dimensions of Standard**} & \multirow{2}{*}{**4**} & \multirow{2}{*}{**5**} & \multirow{2}{*}{**6**} & \multirow{2}{*}{**7**} & \multirow{2}{*}{**8**} \\ \multicolumn{2}{c}{**Normal Distribution**} & & & & \\ \hline \(3\times\)\(m\) & max & \(1.39\times 10^{  
points extracted by MCS are failed, so as to complete the failure-probability calculation of the structure. The training steps of the high-precision BP model proposed in this paper are as follows:  
(1) Extracting key points. For low-dimensional problems (\(m<4\)), firstly, \(10\times m\) groups of key points are extracted from [\(-\)5, 5] by the random integer method, and then the \"*\"-shaped key points are selected. The combination of the two forms the initial sample point set \(X_{i}\). For high-dimensional problems (\(m\geq 4\)), firstly, \(10\times m\) groups of key points are extracted from [\(-\)3, 3] by the random integer method, and then the sample points of \(m\)-dimensional random normal distribution are extracted according to the mean \(\mu_{i}\) and standard deviation \(\sigma_{i}\) of each random variable (the number can be 30 groups). The combination of them constitutes the initial sample point set \(X_{i}\). \(X_{i}\) is substituted into the finite element model for calculation, and the response set \(G(X_{i})\) of structural performance function is obtained. The initial sample point set and the response set together constitute the initial set. (2) Constructing the training set. For low-dimensional problems, the region where the failure surface is located can be determined according to the failure situation of the \"*\"-shaped key points, and then \(0.5\sigma_{i}\) is taken from some key points close to the failure surface to generate 10 groups of sample points, respectively, which are substituted into FEM to calculate the response values. The newly added sample points and the initial set are used together as the training set. For high-dimensional problems, the initial set is directly used as the training set. (3) Calculating the training set's weight and constructing the prediction set. Calculating the weights \(\omega_{i}\) of all sample points in the training set and arranging them in descending order. After the first failure point with \(G<0\) is found, the weight of this point is taken as the boundary, respectively; \(m\) sample points close to 0 with greater weight are found upward, and \((m-1)\) sample points close to 0 with greater weight are found downward. Taking \(0.5\sigma_{i}\) at these sample points to generate 10 groups of sample points to form a prediction set, the number of sample points in the prediction set is \(2m\times 10\). At this time, these sample points do not need to be substituted into FEM for calculation. (4) Training the BP model. Taking the sample point \(X_{i}\) of the training set as input and the response value \(G(X_{i})\) as output, the initial BP model is trained by the MATLAB R2021a Neural Network toolbox, and the response \(G^{\prime}(X_{i})\) of the prediction set is calculated by the BP model. (5) Verifying the model accuracy. The sample points of the prediction set are arranged in ascending order according to the size of \(G^{\prime}(X_{i})\), and then the sample points whose \(G^{\prime}(X_{i})\) is close to 0 are screened out.