## 3 Methodology of Adaptive Gaussian Mixture Model Using Virtual Sample Generation
### Virtual Sample Generation-Based Propagation Strategy  
It is noted that for a black-box model, only sample-based gradient-free methods can be used for single Gaussian component propagation, and fifth-order CR is selected in this paper, as is shown in Equation (19). \[\lambda\int_{\infty}F(\mathbf{x})p(\mathbf{x};\mathbf{m},\mathbf{P})d\mathbf{x}=\int_{\infty}F( \mathbf{A}\mathbf{x}+\mathbf{m})p(\mathbf{x};\mathbf{m},\mathbf{P})d\mathbf{x}\approx\sum_{i=1 }^{N_{p}}W_{i}\mathbf{F}(\gamma_{i}) \tag{19}\]  
where \(\gamma_{i}\big{(}i=1,\cdots,N_{p}\big{)}\) are Cubature points according to the fifth-order CR. Although the Gaussian component splitting technology could reduce unnecessary components, the computation burden, if real samples are involved when propagating, is still intractable. However, the computation burden can be released by using virtual samples with the information hidden in the raw samples (some already obtained real samples):  
\[\int_{\infty}F(\mathbf{x})p(\mathbf{x};\mathbf{m},\mathbf{P})d\mathbf{x}\approx\sum_{i=1}^{N_{ p}}W_{i}\mathbf{F}(\mathbf{\gamma}_{i})\approx\sum_{i=1}^{N_{p}}W_{i}\tilde{ \mathbf{F}}(\mathbf{\gamma}_{i}) \tag{20}\]  
where \(\tilde{\mathbf{F}}(\mathbf{\gamma}_{i})\) is the response of the virtual samples. Generally, virtual samples can be constructed using prior knowledge, the addition of noise, and existing datasets. In this paper, an active sampling-based KRG model based on existing datasets is proposed for generating virtual samples for the black-box uncertainty model. KRG [16; 51] is an interpolation metamodeling technique as shown in Equation (20). \[f_{KRG}(\mathbf{x})=G(\mathbf{x})+Z(\mathbf{x}) \tag{21}\]  
where \(f_{KRG}(\mathbf{x})\) is the output function of the KRG model, \(G(\mathbf{x})\) captures the major trend of input-output mapping, and \(Z(\mathbf{x})\) is a zero-mean, unit-variance stationary Gaussian process that determines the approximation ability of the KRG model. The covariance matrix of \(Z(\mathbf{x})\) is given as follows:  
\[\text{Cov}\big{(}\mathbf{Z}(\mathbf{x}_{i}),\mathbf{Z}(\mathbf{x}_{j})\big{)}=\sigma^{2}\mathbf{R }\big{[}R\big{(}\mathbf{x}_{i},\mathbf{x}_{j})\big{]} \tag{22}\]  
where \(\sigma^{2}\) is the process variance of \(Z(\mathbf{x})\), \(R\) is the correlation matrix, and \(R(\cdot,\cdot)\) is a Gaussian correlation function that controls the smoothness of the KRG model. Note that in addition to the output predicted using the KRG model, the variance, which can be taken as the degree of confidence of the prediction, can also be obtained. The performance of the KRG model will influence the accuracy of the virtual samples, which, in turn, will influence the accuracy of propagation. Although investigations have been conducted to improve the global accuracy of the KRG model, it is still difficult to use a static metamodel to accurately approximate a sophisticated problem (e.g., high-dimensional nonlinear problem) with sparsely scattered samples. However, to improve the convergence and efficiency, the adaptive sampling is actually realized by the management of the online KRG models [46]. In this paper, an adaptive sampling called the \"active sampling method\" is adopted, with which the superior cheap points are adaptively selected. The least-confidence strategy is adopted in this paper to perform the active sampling. The superior sample is selected according to the following metric:  
\[\mathbf{x}^{*}=\arg\max Var_{RBF}(\mathbf{x}) \tag{23}\]  
where \(Var_{RBF}(\mathbf{x})\) is the variance of the output prediction \(f_{KRG}(\mathbf{x})\). Comparison studies in Ref. [46] have proven that the KRG generally has better approximation accuracy than other technologies, for example, RBF and Gaussian Process (GP) [52], in the condition of relatively sparse samples and lack of prior knowledge. In contrast, the construction expense of KRG is more expensive than that of the Radial basis function (RBF) [53].