## 4 Building an Active-Learning-Based Reliability Analysis Platform  
### PCK-Based Active-Learning Method  
The small initial DOE sample size makes it challenging to establish accurate \(K_{\rm b}\) and \(K_{\rm t}\) surrogate models. The output responses of each sample need to be solved by calling the FEM. In order to reduce the number of new samples in the DOE, this study further constructs a reliability analysis platform based on active learning. The purpose of active learning is to establish locally accurate limit state function surrogate models through iteration, thereby reducing the sample size and improving efficiency. The process of active-learning-based reliability analysis and the methods available for each step are shown in Figure 8. The first step is DOE analysis, which has been carried out in the previous section. Further, the metamodels (surrogate models) are built. The next reliability analysis based on the metamodels can be performed by MC, IS, SS, etc. ; in this study, the MC method is used for reliability analysis, and the sample size is chosen to be \(10^{6}\). Then, it is determined whether convergence occurs. If yes, modeling is completed; otherwise, the active learning function is used to find the next optimal point, which is added to the DOE sample set and the metamodel is updated until the convergence condition is met. The sample generation method for the initial DOE can be chosen according to the actual requirements. If the simulation is time-consuming and too many samples are not suitable, LHS sampling can be used. If a large sample size is necessary for the analysis, the MC method can be used. Sobol sampling has the advantage of being able to provide a global sensitivity index, which is suitable for exploring the complex relationship between inputs and outputs. Metamodels used for active learning include kriging, PCK and SVR. In this study, PCK is used to build the metamodel, which combines the strengths of PCE and kriging. The kriging method is a stochastic interpolation algorithm in which the estimated function value is a linear combination of a regression model and a stochastic process. Its general form is [26]  
\[g_{\rm K}(\mathbf{x})=\mathbf{f}^{\rm T}(\mathbf{x})\mathbf{\beta}+\sigma^{2}Z(\mathbf{x},\omega) \tag{2}\]  
where \(\mathbf{f}^{\rm T}(\mathbf{x})\mathbf{\beta}\) denotes the regression model, \(\mathbf{f}(\mathbf{x})=[f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),...,f_{n}(\  
Consider a random vector with independent components \(\mathbf{x}\in\mathbb{R}^{n}\) described by the joint probability density function \(f_{\mathbf{X}}\). PCE approximates the output of the computational model with a sum of orthogonal polynomials [27]:  
\[g_{\text{PCE}}(\mathbf{x})=\sum_{\mathbf{\alpha}\in\mathcal{A}}y_{\mathbf{\alpha}}\Psi_{\mathbf{ \alpha}}(\mathbf{x}) \tag{3}\]  
where \(\Psi_{\mathbf{\alpha}}(\mathbf{x})\) are multivariate polynomials that are orthonormal with respect to the input distributions, \(\mathbf{\alpha}\in\mathcal{A}\subset\mathbb{N}^{n}\) are multi-indices and \(y_{\mathbf{\alpha}}\) are the corresponding coefficients. PCK uses regression-based PCE to capture the global behaviour of the computational model and interpolation-based kriging to capture local variations, and its metamodeling technique is more effective than the metamodeling technique produced by PCE and kriging separately, which is expressed as [28]  
\[g_{\text{PCK}}(\mathbf{x})=\sum_{\mathbf{\alpha}\in\mathcal{A}}y_{\mathbf{\alpha}}\Psi_{ \mathbf{\alpha}}(\mathbf{x})+\sigma^{2}Z(\mathbf{x},\omega) \tag{4}\]  
The active-learning function utilizes the U-function proposed by AK-MC [18]:  
\[U(\mathbf{x})=\left|\frac{\widehat{g}(\mathbf{x})}{\sigma_{\widehat{g}}(\mathbf{x})}\right| \tag{5}\]  
where \(\widehat{g}(\mathbf{x})\) and \(\sigma_{\widehat{g}}(\mathbf{x})\) are the mean and standard deviation obtained from kriging estimation.