## 5 Numerical Example
### Performance of the Hybrid Learning Function  
As an illustrative numerical example in this paper, let us examine a highly nonlinear limit-state function given by  
\[g(\mathbf{x})=(c-1)-x_{2}+\exp(-x_{1}^{2}/10)+(x_{1}/5)^{4} \tag{39}\]  
where \(x_{1}\) and \(x_{2}\) are uncorrelated random variables following normal distribution (\(x_{1}\sim N(2,0.1),x_{2}\sim N(1,0.15)\)), and \(c\) is an integer parameter.  
To compute the failure probability, the proposed framework employs subset simulations and hybrid active learning approaches with \(K_{min}^{max}=3\), and the addition of five DoE points within each enrichment step (\(K=5\)). The performance of Kriging-based reliability analysis is compared using the AK-MCS method [35] with one DoE point enrichment at each iteration. Figure 5 presents the predicted limit-state functions obtained using the proposed framework and AK-MCS during the active learning process. Despite similarities in the predicted limit-state functions between the two methods, it is noteworthy that AK-MCS required more design points for its construction and subsequent reliability analysis. Additionally, Figure 6 illustrates the convergence history of the failure probability (\(c=2\)) computed by subset simulations in the proposed method and MCS in AK-MCS. The results of the failure probability demonstrate reasonable similarity.  
A parametric study is conducted by varying the parameter \(c\) from 3 to 5. The accuracy and efficiency of the different methods (MCS; FORM; SORM; AK-MCS) are compared to the proposed framework in terms of the failure probability \(P_{f}\), the coefficient of variation (c.o.v) of the failure probability \(\Delta_{P_{f}}\), and the generalized reliability index \(\beta\).  
In contrast to FORM, which exhibits less accurate results, and even SORM, which shows significant errors for the case of \(c=4\) due to high nonlinearity, the proposed framework demonstrates consistent accuracy, yielding results akin to the reference value (MCS) for all cases summarized in Table 1. Notably, the performance of the proposed method remains robust even in scenarios with low failure probability, attributed to the effectiveness of active learning and subset simulations. However, it is observed that MCS requires a high number of samples to achieve a certain degree of c.o.v (\(\Delta_{P_{f}}\)). AK-MCS (with \(10^{6}\) samples) was unable to estimate the lower failure probability for \(c=5\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Case** & **Method** & \(\mathbf{P_{f}}\) & \(\Delta_{\mathbf{P_{f}}}\) & \(\mathbf{\beta}\) & \(N_{\mathbf{c}all}\) \\ \hline \multirow{4}{*}{\(c=2\)} & FORM & \(3.61\times 10^{-2}\) & - & 1.798 & 97 \\  & SORM & \(4.03\times 10^{-2}\) & - & 1.747 & 109 \\ \cline{1-1}  & MCS & \(5.14\times 10^{-2}\) & 0.014 & 1.632 & \(10^{5}\) \\ \cline{1-1}  & AK-MCS & \(4.94\times 10^{-2}\) & 0.014 & 1.651 & \(35\,(10^{5})\) \\ \cline{1-1}  & HAK & \(5.15\times 10^{-2}\) & 0.010 & 1.630 & 25 \\ \hline \multirow{4}{*}{\(c=3\)} & FORM & \(3.31\times 10^{-3}\) & - & 2.715 & 145 \\  & SORM & \(2.54\times 10^{-3}\) & - & 2.802 & 157 \\ \cline{1-1}  & MCS & \(4.46\times 10^{-3}\) & 0.03 & 2.615 & \(2\times 10^{5}\) \\ \cline{1-1}  & AK-MCS & \(4.40\times 10^{-3}\) & 0.048 & 2.620 & \(35\,(10^{5})\) \\ \cline{1-1}  & HAK & \(4.51\times 10^{-3}\) & 0.017 & 2.611 & 25 \\ \hline \end{tabular}
\end{table}
Table 1: Comparative analysis of results obtained through the proposed HAK and several other reliability analysis methods for the nonlinear limit-state function.  
Figure 5: Illustration of the limit-state function prediction using multiple DoE points (\(c=2\)). (**a**) Proposed framework with 25 DoE points after three enrichment steps: shapes filled with solid color represent the initial \(K_{min}^{max}\) farthest DoE points identified in each enrichment step; (**b**) AK-MCS with 35 DoE points.  
Figure 6: Convergence history of failure probability (\(c=2\)).