## 1 Introduction
[PERSON] [15] combined SORM with complex-step derivative approximation for sensitivity analysis and design optimization. FORM, as a computationally efficient first-order approximation method, captures linear behavior near the design point but may lose accuracy with strongly nonlinear functions. SORM refines FORM by extending the approximation to second-order statistics, offering a more accurate representation of the limit-state function. While SORM enhances accuracy, it incurs higher computational costs, particularly with an increasing number of input random variables, making it valuable for precise representations of complex limit-state functions. Metamodel-based adaptive methods [16; 17; 18] are a class of techniques used in structural reliability analysis that involve the iterative construction of surrogate models, known as metamodels, which approximate the limit-state function in the immediate vicinity of the limit-state surface. These metamodels are adaptively refined to improve their accuracy, and this refinement process continues until a predefined convergence criterion, typically related to the accuracy of the failure probability, is met. One potent metamodeling approach is polynomial chaos expansion (PCE) [19; 20], which aims to furnish a functional approximation of a computational model by representing it spectrally using a well-constructed basis of polynomial functions. Kriging [16; 21], also known as Gaussian process modeling, is a statistical interpolation technique that harnesses the power of Gaussian processes to interpolate a diverse set of complex functions. It is particularly effective in capturing intricate relationships within data and is widely used in various fields for tasks such as predictive modeling and optimization. [PERSON] and [PERSON] [22] introduced a metamodeling approach, combining nonstationary Kriging and a support vector machine to predict the stochastic eigenvalues of brake systems. In the approach, [PERSON]'s nonstationary kernel was used with step-wise hyperparameters to account for the heterogeneity of the stochastic eigenvalues. In the work by [PERSON] et al. [23], a sampling strategy is introduced to address system reliability-based design optimization problems involving multiple simulation models. Initially, the system uncertainty across the entire domain is characterized through the cumulative distribution function (CDF) of the systems. Subsequently, a new active learning function, known as the generalized expected system improvement, is developed to predict the enhancement in system uncertainty. To address high-dimensional problems using Kriging, [PERSON] et al. [24] employ an adaptive approach to update the surrogate model. This involves sequential exploration-exploitation with a dynamic trade-off in the important dimension or region. The application of the Kriging-based approach for predicting system responses or estimating reliability has found utility in various fields, encompassing geotechnical engineering [25], structural engineering [26], mechanical engineering [27; 28], and medicine [29]. This study primarily focuses on the Kriging modeling approach, considering both homoscedastic noise (constant and uniform variance for all observed responses) and heteroscedastic noise (independent noise with varying variances for each observed response without correlation). The response prediction model for Kriging is defined, and the conditional mean and variance of Kriging predictors are derived. Hyperparameters and the variance of Kriging with noise are determined through maximum likelihood estimation by solving optimization problems. The study introduces an adaptive hybrid learning function, where design of experiment (DoE) points for training a Kriging model are adaptively identified through a hybrid enrichment process.