# An Adaptive Multi-Fidelity Surrogate Model for Uncertainty Propagation Analysis
## 6 Results and Discussion  
The predictive capability of the adaptive Gappy-POD method was investigated first. To achieve this, we evaluated the prediction error of high-fidelity responses on incomplete sample sets. After obtaining high-fidelity responses for all training samples, we obtained the prediction model for uncertainty analysis. The predictive capability of the uncertainty propagation model was then investigated with additional \(n_{\text{test}}\) test samples. Low-speed flow around the NACA0012 airfoil  
In the analysis, \(n_{\text{train}}\) was set to 81, the initial \(n_{\text{complete}}\) to 6, and \(n_{\text{test}}\) to 20. The method used in this paper for selecting high-fidelity samples is divided into two components: initial sample selection and adaptive sampling. Consequently, the results comparison is also conducted separately. Firstly, a comparison of the methods for selecting initial high-fidelity samples was performed without involving adaptive sampling. Here, we primarily compare our clustering-based method using \(k\)-means with the classical random sampling algorithm. The results are presented in section (a). Secondly, a comparison of adaptive sampling algorithms is carried out. We evaluated the adaptive sampling algorithm based on the projection error against the random refinement method. Both methods utilize the same initial complete sample set, which was obtained through the \(k\)-means clustering-based approach. The results are presented in section (b). Finally, a comprehensive comparison of the entire high-fidelity sample selection method was conducted, encompassing both initial sample selection and adaptive sampling. This comparison was made against an entirely random sampling method that includes both initial and adaptive steps. The results are presented in section (c). Comparison of the methods for selecting initial high-fidelity samples  
After acquiring the low-fidelity responses for all training samples, we utilized the \(k\)-means clustering method to classify them into six distinct categories. Figure 7 presents the distribution of friction coefficients and the clustering results for all samples. Notably, significant variations in magnitude can be observed between clusters on the upper and lower surfaces of the airfoil, while the distributions tend to be more concentrated within individual clusters. Consequently, it is reasonable to select a single sample from each cluster to represent the distribution of that particular cluster. Based on the clustering results, we constructed an initial complete sample set and an incomplete sample set. For the complete sample set, high- and low-fidelity responses were obtained through CFD simulations for each sample. For the incomplete sample set, only the low-fidelity responses were computed via CFD simulations (although the high-fidelity responses were also calculated, they were not used in model training but solely for assessing prediction errors). To validate the advantages of our method for selecting initial high-fidelity samples, we compared it with the commonly used random sampling method. In this comparison, the same number of high-fidelity samples was randomly selected from low-fidelity samples. Both methods predicted the high-fidelity responses of incomplete samples using the Gappy-POD method. To eliminate the randomness inherent in random sampling, this comparison was repeated 10 times. The L2 norm errors for both methods are presented in Figure 8. It is evident that the clustering-based approach outperforms random selection, with a median error reduction of over 50% (from 0.0126 to 0.0062) and fewer outliers. The representativeness of initial samples plays a crucial role in model predictive capabilities, and the \(k\)-means clustering method provides a robust foundation for subsequent model refinement by selecting representative initial samples. prediction ability of the Gappy-POD method. Figure 10 presents the evolution of the maximum projection error as the number of adaptive iterations increases.