## 3 Performance Validation of Existing Learning Algorithms
### Validation Results  
Through the above MF data learning method, it is not difficult to find from Figure 5 that the results obtained by using the MF data learning method between the range of 0.55 and 0.75 are closer to the 'True value' than those obtained by the high fidelity data model (HFDM). This is because the amount of HF data is small, leading to the low accuracy of the fitted model. Conversely, the MF data learning method can simultaneously use HF and LF data, and the HF data in the unknown location can be approximated by the known LF data, so the fitting results are relatively good.  
The experiments were repeated one hundred times, randomly utilizing different parameter arrays \(P\). In each model construction, the method of 5-fold cross-validation [107; 108] is used to find out the model with the best parameters, and on this basis, the training and testing are carried out. The average MSE between each MF data learning method and the true value, as well as the average MSE between HFDM and the true value, were calculated.  
Figure 5: The results of one validation among one hundred tests with the parameters in Equation (13). (**a**,**c**,**e**), respectively, show the distribution of multi-fidelity datasets under additive, multiplicative, and mixed noise, where \(X_{l}\) represents low-fidelity data and is represented by blue dots; \(X_{h}\) represents high-fidelity data, represented by a red triangle. (**b**,**d**,**f**) show the curves of three correction methods, true values, and high-fidelity data under corresponding noise, where the true values are represented by blue dashed lines; HFDM represents the high-fidelity data fitting model, in which the fitting method adopts GPR and is represented by the red dotted line; AC, MC, and CC represent additive, multiplicative, and comprehensive correction methods, respectively, and are represented by solid orange, green, and yellow lines, respectively. In addition, the image on the far right is the corresponding enlarged image.  
The results are presented in Table 3. For MF datasets with arbitrary noise, comprehensive correction (CC) has the lowest MSE and the high-fidelity data model (HFDM) has the highest MSE. For MF datasets with only additive noise, the MSE of additive correction (AC) is lower than that of multiplicative correction (MC). For MF datasets with only multiplicative noise, the MSE of MC is lower than that of AC.  
The experimental results show that the results obtained by using the multi-fidelity data learning method are better than those obtained by using only high-fidelity data modeling. Among the three correction methods, the comprehensive correction has the best effect, which is corrected on the basis of additive and multiplicative correction. In addition, we also used RMSE, MAE, \(R^{2}\), and other evaluation methods for testing. We found that the multiplicative correction outperforms the additive correction only in datasets with multiplicative noise. However, in datasets with other types of noise, the multiplicative correction does not outperform the other correction methods. On the contrary, the comprehensive correction consistently yields the best results across all noise types. This further validates the advantage of the comprehensive correction method across various noise datasets. In summary, employing MF data learning methods can enhance model accuracy when faced with limited HF data and abundant LF data. When the noise of multi-fidelity datasets is not known, the comprehensive correction method can obtain better results. The choice of specific correction methods largely depends on the MF dataset. Finally, We also used two statistical test methods, Shapiro-Wilk [109] and Levene [110], to test the mean error (ME) of the four methods under different noises. Readers can find all the relevant codes and results in the data availability statement for this paper.