## 1 Introduction  
The performance and reliability of the aero-engine significantly influence flight safety. As one of the critical components of an aeroengine, the reliability of the turbine casing plays a vital role in ensuring the healthy operation of the aero-engine [1, 2, 3]. The high-pressure turbine casing endures extreme loads and operates in a harsh working environment. Consequently, it is essential to investigate the reliability of the high-pressure turbine casing to enhance the safety and reliability of the aero-engine during aircraft operation. In recent years, numerous studies have been conducted to evaluate the reliability and service life of turbine components. In this context, the reliability analysis of turbine components remains a challenging task, given the complexity of their structures, the high dimensionality of input variables, and the strong nonlinearity of responses under severe operating conditions. In recent years, numerous studies have been conducted to evaluate the reliability and service life of turbine components. For instance, [PERSON] et al. [4] proposed a multi-layered method for assessing steam turbine life, integrating damage calculation, probabilistic analysis, and fracture mechanics. [PERSON] et al. [5] developed a methodology to extend the lifespan of steam turbine rotors by utilizing ultrasonic inspection data for fatigue life assessment while incorporating uncertainties through probabilistic approaches. Despite their contributions, these methods face significant challenges, such as high computational complexity, strong dependency on extensive data, and limited generalizability of the models, particularly for complex structural components in aero-engines. Therefore, efficient and accurate methods are urgently needed to address these limitations and improve the reliability analysis of turbine components. Structural reliability analysis is based on probability theory, mathematical statistics, and stochastic process theory, which assesses the probability that a structural system can withstand applied loads and environmental impacts while maintaining its functionality within a specified service life under given working conditions. The limit state function of a structure is commonly denoted as G, where G < 0 indicates structural failure. The failure probability can be approximated through numerical methods and is expressed as the frequency of failure sample points among all sample points [6]. The analysis methods primarily consist of numerical methods and approximation methods. Among the numerical methods, the Monte Carlo simulation (MCS) [7] is a widely recognized sampling-based approach that provides high accuracy and reliability in estimating failure probabilities. Failure probability is represented by the ratio of failure sample points to the total number of sample points. Despite its accuracy, MCS becomes computationally prohibitive when large-scale simulations are required, particularly when finite element model calculations are involved. Therefore, variance reduction techniques such as importance sampling (IS) [8] have been proposed to reduce the number of required samples. Nevertheless, IS still demands tens of thousands of sample points to obtain reliable results. Another approach involves employing surrogate model techniques to reduce the computational cost associated with individual sample points in MCS. [PERSON] et al. [9] proposed an improved MCS method that distributes sample points uniformly in the design space, using the probability density value of sample points as weights to approximate the most probable failure points (MPP) and identify the failure region. This method significantly reduces computational cost while maintaining accuracy, although it still relies on traditional approximate methods for identifying the MPP and failure region. [PERSON] et al. [10] introduced a sampling method for model inference and uncertainty quantification, enabling the estimation of parameter distributions and prediction uncertainties.