## 1 Introduction  
The aim of the structural reliability analysis is to meticulously quantify the failure probability that arises from a multitude of uncertain factors, such as the geometric size error, variations in material properties, and external applied load [1; 2]. To improve system reliability and simplify the reliability-based design problems widely applied in various engineering practices [3; 4], many scholars have studied the reliability of multi-failure mode systems [5; 6; 7; 8; 9]. For example, [PERSON] et al. [7] conducted a reliability assessment to solve specific safety assessment problems, analyzed the underlying causes of micro-electro-mechanical system device failure, and carried out a failure test using a typical micro-electro-mechanical system device package structure. [PERSON] et al. [8] introduced a probabilistic analysis framework aimed at evaluating the reliability of turbine discs while accounting for the interdependencies among various failure modes. [PERSON] et al. [9] proposed a hybrid reliability-based topology optimization method to deal with cognitive and cognitive uncertainty. [PERSON] et al. [10] proposed an innovative reliability analysis method that integrates Markov chain Monte Carlo method with random forest algorithms. [PERSON], [PERSON], et al. [11] explored the properties of steel by evaluating the seismic reliability of deep column smf. The first-order reliability method, response surface method, and an advanced probability scheme were adopted. The efficiency and accuracy of the proposed method were verified by the traditional Monte Carlo simulation method. German [PERSON], [PERSON], [PERSON], et al. [12] evaluated the Usumacinta Bridge in Mexicobased on probabilistic methods, The Sentinel-1 image was used to define structural reliability. The case for selecting probabilistic methods integrates the reliability of the structure. Displacement theory and probability density function obtained by InSAR technique (pdf). Common structural reliability analysis methods include the digital simulation method [13; 14], approximate analytical method [15; 16] and surrogate model methods. The Monte Carlo method [17] is a relatively straightforward digital simulation method. However, the large number of samples required for obtaining an accurate solution makes the method less efficient. Another digital simulation method known as importance sampling [18] utilizes the design point of the limit state equation as its sampling center, thereby enhancing sampling efficiency to a certain extent. However, this design point must be predetermined, which limits the applicability of this method to complex structures. Approximate analytic methods, such as the first-order second-moment method [19], necessitate minimal computational effort but yield large relative errors. Consequently, these methods are only suitable for solving the reliability problem associated with an explicit limit state equation. In view of the nonlinear implicit characteristics of complex mechanical structures, surrogate modeling methods such as the Kriging algorithm [20], neural network algorithm [21], and support vector machines algorithm [22] are frequently utilized to assess the reliability of these intricate mechanical structures. However, these algorithms are simple extensions of the single traditional classification algorithm for specific data, and they inevitably inherit the inherent shortcomings of the original algorithm [23]. Simultaneously, these algorithms rely on relatively idealized assumptions regarding probability distribution and data types when dealing with data uncertainty, which consequently hampers their widespread applicability in practical projects [24]. Traditional reliability analysis methods have accuracy problems, large errors, and low computational efficiency, making it difficult to effectively apply them in practical engineering.