## 3 Performance Validation of Existing Learning Algorithms
### Multi-Fidelity Data Acquisition and Processing  
In this validation, MF datasets are artificially constructed to obtain datasets. Here, we assumed an ideal function as Equation (12) to compute the true values.  
\[Y_{true}(x)=p_{1}*sin\left(p_{2}*x^{2}+p_{3}\right)*\left(p_{4}*x^{2}+p_{5} \right)+p_{6}*\left(p_{7}*x^{3}+p_{8}\right)+p_{9}*x+p_{10} \tag{12}\]  
where \(p_{1}\!\sim\!p_{10}\) are random parameters. We conducted a total of one hundred tests to validate our approach. For brevity, we present the details of one with parameters (inEquation (12)) set as \(P_{sample}\) to typify the behavior observed across one hundred repetitions. The parameters selected are as follows:  
\[P_{sample}=[-1,\ -5,\ -4,\ 5,\ -1,\ -2,\ 1,\ 0,\ -4,\ -2] \tag{13}\]  
where the true value function corresponding to the parameter \(P_{sample}\) is shown in Figure 4. We selected the results of one of these for display, as shown in Figure 5, which includes the distribution of HF and LF data points, true value, HF data fitting model, and different correction methods under different noises.  
In a certain range, this function can exhibit wave-like behavior. In reality, we can assume that the function curve is obtained from a plane wave. Based on this assumption, we can add additive noise, multiplicative noise, and mixed noise to the function for different MF data learning methods. The LF data input value of the validation was two hundred points equidistant from zero to one, denoted by \(X_{l}\). The input values for the HF data are contained in the LF data input values, which are twenty equidistant points between zero and one, denoted by \(X_{h}\). The response values of HF and LF data are represented by \(Y_{h}\) and \(Y_{l}\), respectively. The equation is as follows:  
\[Y_{h}=E_{h_{+}}*Y_{true}+E_{h_{+}} \tag{14}\]  
\[Y_{l}=E_{l_{+}}*Y_{true}+E_{l_{+}} \tag{15}\]  
where \(E_{h_{+}}\) and \(E_{l_{+}}\), respectively, represent multiplicative noise under HF and LF; and \(E_{h_{+}}\) and \(E_{l_{+}}\), respectively, represent additive noise under HF and LF. The additive noise in the HF data is relatively small, while the additive noise in the LF data is relatively large. The multiplicative noise in the HF data is close to one, while the multiplicative noise in the LF data is far from one. The above formula is the construction formula for the MF data in the comprehensive correction method. In Equations (14) and (15), we can set parameters \(E_{h_{+}}\) and \(E_{l_{+}}\) to one for generating datasets only containing additive noise and set parameters \(E_{h_{+}}\) and \(E_{l_{+}}\) to zero to generate datasets with multiplicative noise only. During validation, for the datasets with additive noise, random numbers ranging from 0 to 1 are generated. These numbers are then multiplied by 0.4 and 0.8 to obtain the values for \(E_{h_{+}}\) and \(E_{l_{+}}\). For multiplicative noise, we randomly select numbers within the range of 0.9 to 1.1 to represent \(E_{h_{+}}\), and within the range of 0.8 to 1.3 to represent \(E_{l_{+}}\), respectively.  
The above HF and LF datasets with different noise added are randomly shuffled as the training set, while the input values of the test set were an additional twenty points from zero to one that were not included in the HF and LF datasets. The mean square error (MSE)  
Figure 4: All true value functions. (**a**) one hundred functions of true value in the performance validation. The bolded one is the function with the parameters in Equation (13). (**b**) enlarged plot of the true value function shown.  
was used as an evaluation method, and the experiment was repeated one hundred times randomly with different parameter arrays \(P\) to calculate the mean MSE between each MF data learning method and the true value and the mean MSE between the high-fidelity data model (HFDM) and the true value.