## 2 Adam-Bayesian-GPR-Based MC Reliability Calculation Procedure Framework
### Combined Kernel Function Construction and Selection  
GPR defines the homogeneity difference between different observation data as a kernel function, which not only defines a general kernel function but also allows us to customize the kernel function according to different problems. The selection of a kernel function is the key to ensuring the accuracy of GPR.  
The GPR-based reliability analysis largely depends on the selected covariance function, that is, the quality of the kernel function. The basic idea of the kernel function is to map the undivided data in the low-dimensional space to the high-dimensional space to achieve effective separation. In this study, we investigate the influence of different commonly used kernel functions on the calculation results of GPR and build combined kernel functions for more accurate and effective thinking and learning.  
Because there is no periodic regression trend in the data in this study, the following basic kernel functions are selected from common kernel functions of GPR [30]: i. radial basis kernel function (_RBF_) (see Formula (8)); ii. Matern kernel function denoted by Formula (9); iii. Matern kernel function of \(v=1.5\) (_K1.5_, see Formula (10)); iv. Matern kernel function of \(v=2.5\) (_K2.5_, see Formula (11)).  
\[K_{\text{RB}}\big{(}x,x^{\prime}\big{)}=\exp\!\left(-\frac{\|d\|^{2}}{2l^{2}}\right) \tag{8}\]\[K_{Matern}\big{(}x,x^{\prime}\big{)}=\frac{2^{1-v}}{\Gamma(v)}\Bigg{(}\frac{\sqrt{2v }|d|}{l}\Bigg{)}^{v}K_{v}\Bigg{(}\frac{\sqrt{2v}|d|}{l}\Bigg{)} \tag{9}\]  
\[K_{\text{Matern},15}\big{(}x,x^{\prime}\big{)}=\Bigg{(}1+\frac{\sqrt{3}|d|}{l} \Bigg{)}\exp\Bigg{(}-\frac{\sqrt{3}|d|}{l}\Bigg{)} \tag{10}\]  
\[K_{\text{Matern},\,2.5}\big{(}x,x^{\prime}\big{)}=\Bigg{(}1+\frac{\sqrt{5}|d|}{ l}+\frac{5|d|^{2}}{3l^{2}}\Bigg{)}\exp\Bigg{(}-\frac{\sqrt{5}|d|}{l}\Bigg{)} \tag{11}\]  
where \(x\), \(x^{\prime}\) are the input data of the training sample and the input data of the test sample, respectively; \(d\) is Euclidean distance; \(l\) is the length scaling factor, which controls the fluctuation degree, and its default value is \(1.0\); \(v\) is the parameter used to control the smoothness of the resulting function, and the common value in the field of machine learning is \(v=1.5\) or \(2.5\); \(\Gamma(\cdot)\) is the gamma function; \(K_{\text{v}}(\cdot)\) is the modified Bessel function of the second kind [31].  
Under the sufficient condition that [PERSON]'s theorem of kernel functions is satisfied, the kernel operations of addition and multiplication can be carried out for the above classical kernel functions (see Formulas (8)-(11)), and then the data can be modelled based on the new combined kernel. In order to select the kernel function with the best surrogate model performance, we study the performances of RBF, K1.5, K2.5, and the following combined kernel functions: \(RBF+K_{\text{Matern},15}(x,x^{\prime})\) (\(RBFK1.5\)); \(RBF+K_{\text{Matern},\,2.5}(x,x^{\prime})\) (\(RBFK2.5\)); Constant \(\times RBF\) (see Formula (12)) (\(CRBF\)), based on nearly a hundred trial calculations.  
\[K_{\text{RBC}}\big{(}x,x^{\prime}\big{)}=\text{C}\cdot\exp\Bigg{(}-\frac{\|d \|^{2}}{2l^{2}}\Bigg{)} \tag{12}\]  
where C is constant.