## 4 Framework for Reliability Analysis Incorporating Noisy Response Prediction
### Adaptive Hybrid Learning Function
After the \(K-K_{min}^{max}\) number of DoE points is determined by Equation (32), then candidate points \(\hat{\mathbf{x}}\in\underline{\mathcal{Q}}\) for DoE are employed to compute weights according to:  
\[\text{weight}:w(\hat{\mathbf{x}})=\Phi\Bigg{(}\Bigg{(}\frac{\big{|}\mu_{ \tilde{G}}(\hat{\mathbf{x}})\big{|}}{\sigma_{\tilde{G}}(\hat{\mathbf{x}})} \Bigg{)}^{2}\Bigg{)} \tag{34}\]Using the computed weights, the \((K-K_{min}^{max})\) number of weighted centroids of clusters denoted by \(\mathbf{w}\mathbf{x}_{c}=\left\{\hat{w}x_{c_{1}},\hat{w}x_{c_{2}},...,\hat{w}x_{c _{K-K_{min}}^{max}}\right\}\) is determined by the \(K\)-means clustering algorithm. Subsequently, a point \(\hat{x}\in\mathcal{Q}\) that is closest to each weighted centroid is chosen as a point in the DoE. In the proposed hybrid learning process, candidate points are initially generated in addition to the \(n_{0}\) initial DoE points. From this pool of candidates, points falling within the \(\gamma\)-quantile of \(\left|\mu_{\hat{G}}(d)\right|\) are selected. These chosen points are then utilized to calculate individual distances to each point in the initial DoE. Subsequently, the point with the farthest distance is determined as a DoE point and added to the initial set. This iterative process continues until the specified \(K_{min}^{max}\) is met, with the recommended setting being a half or one-third of the number of points, denoted as \(K\), to add within an enrichment step. Following this, \(K\)-means clustering algorithms are employed to identify the \(K-K_{min}^{max}\) number of weights and centroids of points within a subset of the candidate pool \(\mathcal{Q}\). The next step involves selecting \(K-K_{min}^{max}\) points closest to the computed centroids of clusters, which are then added to the DoE. In summary, the proposed hybrid learning function initially identifies points near the limit-state approximation that are farthest from existing training points, followed by an exploration of the region expected to be close to the limit-state surface. This bidirectional process ensures that the newly added DoE points, upon completion of the enrichment process, include those distant from the existing DoE to explore unknown regions. Simultaneously, the search is directed toward samples relatively close to the limit-state function, significantly impacting the failure probability. Figure 3 depicts the hybrid learning process for the limit-state function \(g(x_{1},x_{2})=5-2x_{1}+0.2x_{2}^{3}\), where \(x_{1}\) and \(x_{2}\) represent uncorrelated standard normal random variables. Figure 3: Illustration of the proposed hybrid active learning process with the initial five design of experiment (DoE) points (\(K_{min}^{max}=3\), \(n_{0}=5\), the number of points to be added to DoE \(K\)= 5, \(\gamma=0.1\)). Enrichment process (**a**) initial Kriging and candidate points within \(\gamma\)-quantile of \(\left|\mu_{\hat{G}}(d)\right|\) and a selected point (\(\hat{x}^{*}\)) designated for addition as a DoE point, positioned as the farthest from the existing DoE points, (**b**) the second \(\hat{x}^{*}\) (the Kriging model is updated with the first \(\hat{x}^{*}\)), (**c**) the third \(\hat{x}^{*}\), (**d**) updated Kriging model and candidate points for \(K\)-means clustering, centroids of clusters, and determined points based on centroids to be added to the DoE.