## 4 Framework for Reliability Analysis Incorporating Noisy Response Prediction
### Adaptive Hybrid Learning Function  
Constructing an accurate Kriging model requires a sufficiently large number of DoE points. However, this necessity often leads to significant computational costs, especially when the original function's evaluation is computationally expensive. To address thischallenge and facilitate effective enrichment in DoE, recent years have witnessed the development of various adaptive methods [16; 35; 36; 37]. These methods employ a sequential point addition approach based on information from previous iterations. In essence, adaptive methods iteratively construct Kriging models that approximate the limit-state function in close proximity to the limit-state surface. These models undergo adaptive refinement through the inclusion of additional limit-state function evaluations in their experimental designs until a convergence criterion, tied to the accuracy of the probability of failure, is satisfied. The significance of the misclassification error within the adaptive refinement stages of the Kriging surrogate model hinges on the relative importance of the associated locations. In reliability scenarios, domains critical to accurate failure probability and reliability index estimation, particularly those containing the design point, assume paramount importance. Consequently, adaptive refinement strategies are inclined to prioritize these critical domains. Specifically, a learning function is employed to identify points that, upon integration into the experimental design, can most effectively reduce the error introduced by the surrogate in estimating the failure probability. A diverse array of learning functions has been put forth in the literature [38; 39]. For example, [PERSON] et al. [35] suggested the integration of Kriging and MCS in a method known as AK-MCS. This approach initially creates Monte Carlo sample points in accordance with the PDF. The evaluation of the significance of each Monte Carlo sample relies on past simulation outcomes, with the learning function being explicitly defined as  
\[U_{LF}(x)=\frac{\big{|}\mu_{\tilde{G}}(x)\big{|}}{\sigma_{\tilde{G}}(x)} \tag{31}\]  
where \(\mu_{\tilde{G}}(x)\) signifies the Kriging mean prediction and \(\sigma_{\tilde{G}}(x)\) represents the associated standard deviation. In AK-MCS, the next simulation point is determined by selecting the Monte Carlo sample that minimizes the learning function in Equation (31) from the candidate pool \(\mathcal{Q}\). In this study, a hybrid active learning process is introduced to identify multiple points in the DoE during each enrichment process. This learning function integrates the DoE max-min function by [PERSON] and [PERSON] [40] with a \(K\)-means clustering algorithm [41]. The max-min learning function seeks to identify samples in the vicinity of the limit-state approximation that are farthest from existing training points. The next point to be added to the DoE is determined by  
\[\hat{x}^{*}=\arg\max_{d\in\mathcal{Q}^{\prime}}\min_{i=1,\ldots,N}\Big{\|}d-x^{ (i)}\Big{\|},\ \mathcal{Q}^{\prime}=\big{\{}d\in\mathcal{Q}:\big{|}\mu_{\tilde{G}}(d) \big{|}<\gamma\text{-quantile of }\big{|}\mu_{\tilde{G}}(d)\big{|}\big{\}} \tag{32}\]  
In adherence to the min-max learning function approach, enrichment occurs sequentially, adding one DoE point at a time, and updating the min-max distance after the inclusion of each point. To identify the multiple \(K\) number of DoE points in each iterative enrichment process, the \(K\)-means clustering algorithm can be employed [42]. The proposed framework integrates the weighted \(K\)-means clustering algorithm. \(K\)-means clustering begins by defining a subset of the candidate pool \(\mathcal{Q}\) as  
\[\underline{\mathcal{Q}}=\big{\{}\mathbf{x}\in\mathcal{Q}:-\alpha\cdot\sigma_{ \tilde{G}}(\mathbf{x})\leq\mu_{\tilde{G}}(\mathbf{x})\leq\alpha\cdot\sigma_{ \tilde{G}}(\mathbf{x})\big{\}} \tag{33}\]  
where \(\alpha\) defines the confidence intervals.