# An Improved Monte Carlo Reliability Analysis Method Based on BP Neural Network
## 1 Introduction  
In engineering structures such as bridges, tunnels, and slopes, there are several uncertain parameters, including external loads, mechanical properties of materials, and structural dimensions. These factors introduce significant uncertainty in the structural performance, which may lead to premature failure and make it difficult to achieve the expected service life. Structural reliability refers to the probability that an engineering structure will fulfill its intended function under specified time and service conditions, generally measured by the failure probability of the structure. There are various approximate methods for calculating structural failure probability, such as the first-order reliability method (FORM) [1], the second-order reliability method (SORM) [2], the response surface method (RSM) [3], importance sampling (IS) [4], and Monte Carlo simulation (MCS) [5]. MCS is an analysis method based on mathematical statistics, which defines the frequency of failure sample points in all sample points as failure probability. When the number of sample points is sufficiently large, the results exhibit high confidence [6]. However, when the calculation time of a single sample point is long (such as finite element calculation) or the failure probability is low,resulting in a large number of samples required, the huge number of sample points will greatly increase the time cost, reducing the applicability of this method. In order to solve the problem of high time cost of the Monte Carlo reliability analysis method when performing a large number of sample point calculations, many scholars have proposed different methods for calculating surrogate models. The general approach is to replace the time-consuming finite element model (FEM) with an approximate model. Commonly used approximate models include polynomial response surface [7], kriging model [8], support vector machine (SVM) [9], and artificial neural network (ANN) [10,11], among others. The polynomial response surface method is simple and uses an explicit structural performance function, leading to short computation times, but it is only suitable for weakly nonlinear cases. The kriging model offers high accuracy, but its large initial sample size often requires acceleration methods for optimization [12, 13, 14]. [PERSON] et al. proposed AK-MCS to reduce the calculation times of the finite element model while ensuring accuracy, and proposed AK-MCS-U, AK-MCS-EFF, and other active learning methods to accelerate [15]. With the increasing application of machine learning algorithms in civil engineering, implicit structural performance functions constructed by intelligent algorithms (such as SVM [16,17], ANN [18,19], BPNN [20], etc.) have also been applied to structural reliability analysis. These intelligent algorithms possess strong nonlinear fitting and generalization capabilities [21], which result in higher accuracy for the prediction models they generate. Consequently, the MCS failure probability results are more precise, though the computation time is longer than that of methods using explicit structural performance functions. Comprehensively comparing the scope of application and limitations of different surrogate models, this paper adopts the machine learning algorithm with higher prediction accuracy for the construction of the surrogate model in the MCS method, but the calculation time of the implicit structural performance function formed by the surrogate model is longer, and how to cut down the number of calculations of the surrogate model is to be explored. While ensuring the accuracy of the failure probability calculation, how to efficiently select sample points that can represent the global design space is the key to reduce the calculation time. [PERSON] et al. took the probability density value of sample points as the weight, proposed to distribute sample points uniformly in the design space, and calculate the failure probability by approaching the most likely failure point (MPP) [22].