# An Improved Monte Carlo Reliability Analysis Method Based on BP Neural Network
## 5 Conclusions  
Aiming at engineering structures with complex models, long time-consuming single-FEM calculations and low failure probabilities, this paper proposes an improved Monte Carlo reliability analysis method (Improved BP-MCS) based on the BP neural network algorithm. By calculating the failure probabilities of four explicit mathematical models and one implicit finite element model, the accuracy, efficiency, and engineering applicability of the Improved BP-MCS are verified. The specific conclusions are as follows:  
(1) Taking the \(m\)-dimensional standard normal distribution as the research object, \(10\times m\) groups of key points were extracted by means of the random integer method. It was found that the weight distribution was relatively uniform, and its minimum order of magnitude could reach that when \(10^{7}\) sample points were extracted using Monte Carlo Simulation (MCS). Moreover, when combined with the key points generated from the random sampling of variables, the global design space could be uniformly covered. (2) Five numerical examples are used to compare the Improved BP-MCS with some accelerated algorithms. The results indicated that the relative error of the failure probability calculated by the Improved BP-MCS was within 2%. There was little difference in the calculation times of the finite element model, yet the number of invocations of the surrogate model was significantly reduced. The Improved BP-MCS algorithm can be applied to reliability calculate problems of different dimensions. (3) Comparing the failure probability calculation time of Direct BP-MCS and Improved BP-MCS, the results demonstrated that when the failure probability was low, the Improved \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & \(N_{FEM}\) & \(N_{call}\) & \(P_{f}\) & \(COV_{P_{f}}\) & \(\epsilon\) \\ \hline MCS [28] & \(5\times 10^{6}\) & \(5\times 10^{6}\) & 7.402 \(\times\) 10\({}^{-4}\) & 0.0164 & \\ AK-MCS-U [36] & 91 & \(5\times 10^{6}\) & 7.417 \(\times\) 10\({}^{-4}\) & 0.0164 & 0.2\% \\ AK-MCS-IRS [28] & 29 & \(5\times 10^{6}\) & 7.300 \(\times\) 10\({}^{-4}\) & 0.0165 & 1.38\% \\ Direct BP-MCS & 93 & \(5\times 10^{6}\) & 7.334 \(\times\) 10\({}^{-4}\) & 0.0165 & 0.92\% \\ Improved BP-MCS & 93 & 122,767 & 7.350 \(\times\) 10\({}^{-4}\) & 0.0165 & 0.7\% \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Reliability calculation results of three-unequal-span continuous girder. BP-MCS required fewer invocations of the BP model, which could substantially reduce the calculation time. The sampling strategy of random integer method combined with random sampling of variables proposed in this paper can effectively improve the computational accuracy of the surrogate model under fewer finite element model calculations, while the improved Monte Carlo reliability analysis method can effectively reduce the number of surrogate model calls in the calculation of failure probability. It should be noted that the method in this paper has high accuracy and computational efficiency in the reliability problem in high dimensions, but the sampling strategy is different in low dimensions, and the number of finite element model calculations required for constructing the surrogate model is at a medium level, so the unity of the computational method from different dimensions can be considered for subsequent research. Meanwhile, the performance of the Improved BP-MCS algorithm proposed in this paper is based on the nonlinear prediction ability and generalization ability of the BP neural network algorithm, because the BP neural network algorithm has the characteristics that may fall into the local optimal solution and has the initial parameter dependence. Therefore, the subsequent research can be adjusted to find the optimization of the initial parameters of the BP model through some intelligent optimization algorithms such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). Conceptualization, X.S. ; Methodology, L.Z. ; Software, L.Z. ; Validation, M.T. ; Resources, X.S. ; Data curation, M.T.