## 1 Introduction
However, this approach still evaluates a large number of sample points in the non-failure region, resulting in low computational efficiency despite its high accuracy. To address the limitations inherent in numerical methods, approximate methods have been extensively employed in structural reliability. The fundamental principle of these methods lies in substituting the original limit state function with an approximate representation, thereby significantly enhancing computational efficiency. Among these methods, the first-order second moment (FOSM) method [11] and the second-order reliability method (SORM) [12] estimate failure probabilities by expanding the limit state function approximately. Despite their widespread use, these methods face notable challenges when applied to problems involving multiple most probable failure points or pronounced nonlinearity in the vicinity of the failure surface. To overcome such challenges, surrogate models have emerged as an alternative. Commonly used surrogate models include the Kriging model [13; 14], polynomial chaos expansion (PCE) [15; 16], and response surface methods (RSM) [17; 18], which have demonstrated effective performance in addressing these issues. The Kriging model is renowned for its strong local interpolation properties, enabling the prediction of both response values and associated errors at sample points. Leveraging this capability, numerous adaptive Kriging models [19; 20; 21] have been developed for structural reliability analysis. However, the construction of the Kriging model is computationally intensive for large datasets, and its performance deteriorates in high-dimensional problems. Similarly, the PCE model exhibits strong global interpolation properties but suffers from the \"curse of dimensionality\", where computational costs increase exponentially with the number of input variables. To mitigate this, sparse approximation techniques have been proposed, but challenges remain for high-dimensional problems. The RSM is comparatively simpler to construct but frequently lacks accuracy when dealing with complex limit state functions. Consequently, researchers have developed advanced versions of the RSM, such as high-order response surface methods and continuously interpolated high-precision response surface methods. Despite these advancements, the quality and accuracy of the surrogate model largely depend on the sample data, posing challenges in addressing multi-dimensional and high-dimensional problems. While these approaches improve efficiency, they face limitations such as reduced accuracy in highly nonlinear problems, the curse of dimensionality, and high reliance on sample data quality. With the rapid development of machine learning, the neural network-based surrogate model has demonstrated significant potential in structural reliability analysis. Unlike traditional methods, the neural network model avoids the need to construct complex explicit limit state functions and offers higher accuracy and faster computational speed, particularly when addressing high-dimensional and highly nonlinear problems. Common neural network approaches include the artificial neural network (ANN) [22; 23; 24] and back propagation neural network (BPNN) [25; 26; 27]. The ANN model exhibits powerful nonlinear modeling capabilities and adaptability, but its high data requirements and expensive training costs often limit its applicability in certain scenarios. As a significant branch of the ANN, the BPNN model features efficient learning capabilities. By employing an error back propagation algorithm, the BPNN iteratively adjusts weights to minimize errors, thereby optimizing network performance and improving learning efficiency. However, BPNNs face challenges such as susceptibility to local optima, reliance on gradient information, and slow convergence speeds, which can limit their accuracy and efficiency in certain applications. To address these challenges, Particle Swarm Optimization (PSO) has been proposed as an effective optimization technique.