## 2 Multi-Fidelity Data Learning Methods
### Comprehensive Correction  
When the HF response value cannot be approximated by a simple additive correction or multiplicative correction, it is better to consider using a comprehensive correction (CC), which integrates additive correction and multiplicative correction. Generally, it can be expressed as Equation (3):  
\[\hat{y}_{h}(x)=\rho(x)y_{l}(x)+\delta(x) \tag{3}\]  
where \(\rho(x)\) and \(\delta(x)\) are the multiplicative correction surrogate model and the additive correction surrogate model, respectively, and \(\rho(x)\) is usually replaced by a constant \(\rho\)[9; 90; 95; 101]. The replacement equation is as follows:  
\[\hat{y}_{h}(x)=\rho y_{l}(x)+\delta(x) \tag{4}\]  
In the validation section of this paper, Gaussian process regression (GPR) [102; 103] is employed as the surrogate model. GPR is based on Bayesian principles, wherein it characterizes model uncertainty by constructing a prior distribution and updates this distribution based on the observed data to obtain a posterior distribution. Specifically, GPR utilizes a kernel function to compute the correlations between the data and subsequently constructs a covariance matrix. The procedure is as follows:  
Assuming we have a dataset \((X,y)\), where \(X\) and \(y\) represent the input features of the data and their corresponding labels, and we have new data \(x_{N^{\prime}}\) that we want to predict using GPR, GPR constructs a covariance matrix \(K\) that describes the relationships among all the data (including the white noise):  
\[K= \begin{bmatrix}K_{N}+\sigma_{n}^{2}I&K_{NN^{\prime}}\\ K_{N^{\prime}N}&K_{N^{\prime}N^{\prime}}\end{bmatrix}  \tag{5}\]  
where \(K_{N}\) represents the covariance matrix of the original data \(X\), \(K_{NN^{\prime}}\) and \(K_{N^{\prime}N}\) represent the covariance matrices between the data points \(X\) and the data point to be predicted, and \(K_{N^{\prime}N^{\prime}}\) represents the covariance matrix of the data point to be predicted. \(\sigma_{n}^{2}\) represents the variance of the white noise added during modeling, and \(I\) denotes the identity matrix. For each element within the covariance matrix, it can be computed using a covariance function; in this case, we employ the Gaussian kernel function (also known as the radial basis function):  
\[K^{ij}=k(x,x^{\prime})=\sigma_{f}^{2}\exp\left(-\frac{1}{2\sigma_{l}^{2}}\|x- x^{\prime}\|^{2}\right)+\delta_{ij}\sigma_{n}^{2} \tag{6}\]  
where \(\sigma_{f}\) and \(\sigma_{l}\) are hyperparameters that control the characteristics of the covariance function. \(\delta_{ij}\) is used to ensure that only elements where \(i=j\) are affected by the noise term, representing that the observation errors between different data points are uncorrelated. \(\|x-x^{\prime}\|\) represents the Euclidean distance between two data points. Once the estimation of model hyperparameters is completed, the mean \((\mu_{N^{\prime}})\) and variance \((\Sigma_{N^{\prime}})\) of the prediction result are obtained by maximizing the conditional likelihood:  
\[\mu_{N^{\prime}}=K_{N^{\prime}N}(K_{N}+\sigma_{n}^{2}I)^{-1}y \tag{7}\]  
\[\Sigma_{N^{\prime}}=K_{N^{\prime}N^{\prime}}-K_{N^{\prime}N}(K_{N}+\sigma_{n}^ {2}I)^{-1}K_{NN^{\prime}}+\sigma_{n}^{2}I \tag{8}\]  
In practical applications, the prediction result is often represented by the mean \((\mu_{N^{\prime}})\). In the correction methods used in this paper's validation, GPR is employed as the surrogate model. Specifically, for the comprehensive correction, a constant \(\rho\) is used to replace the multiplicative correction term, and the equation is as follows:  
\[\hat{y}_{h}(x)=\rho\hat{y}_{l}(x)+\delta(x) \tag{9}\]  
where \(\hat{y}_{l}(x)\) and \(\delta(x)\) represent surrogate models for the LF data and discrepancy data, respectively. The value of \(\rho\) can be obtained by minimizing the following equation:  
\[\min:\left(\delta(x_{h})-d_{h}\right)^{2} \tag{10}\]  
\[d_{h}=\rho\hat{y}_{l}(x_{h})-y_{h} \tag{11}\]  
The solution for \(\rho\) can be determined through the method of least squares to minimize Equation (10).