# An Adaptive Multi-Fidelity Surrogate Model for Uncertainty Propagation Analysis
## 6 Results and Discussion
After four iterations, the maximum projection error has decreased to a level below the set threshold of 0.5%, indicating rapid improvement in the accuracy of the adaptive method. Next, we compared the prediction errors of the Gappy-POD model using the adaptive sampling algorithm based on projection errors and the random sampling method. Both methods are based on the same initial high-fidelity sample set, which was selected through the \(k\)-means clustering method. Figure 11 presents a box plot of the prediction error for two methods, showing the average error and the 99% confidence interval. With the same number of complete samples, the mean prediction error of the adaptive method is consistently smaller, and its 0.995 quantile value also significantly decreases. The advantage of the adaptive sampling algorithm based on low-fidelity projection errors lies in its ability to more selectively incorporate samples that are poorly predicted by the Gappy-POD method into the training set, thereby effectively improving the  
times to eliminate the influence of randomness. As the training sample size increases, the statistical results of prediction errors are shown in Figure 12, representing the average error and the 99% confidence interval. Under the same number of complete samples, the proposed high-fidelity sample experimental design method consistently achieves a smaller mean error than the random sampling method. For example, after 10 adaptive refinement iterations, the mean error of the proposed method is approximately 30% lower than that of the random method (0.00288 vs. 0.00400). Additionally, the 0.995 quantile value of the error significantly decreases. This indicates that the experimental design method not only improves overall prediction accuracy, but also reduces the occurrence of larger errors. Figure 13 presents a comparison between the wall friction coefficient distribution predicted by the Gappy-POD method and the high-fidelity CFD simulation on a randomly selected incomplete sample. It is evident that the two predictions are in close agreement, indicating minimal differences. Given that low-fidelity CFD simulations are significantly less costly than high-fidelity simulations, we can eliminate the need for time-consuming high-fidelity CFD simulations. Figure 11: Box plot of the prediction error with adaptive and random sampling methods. Figure 12: Box plot of the prediction error with the entire experimental design algorithm and random sampling method (NACA0012 case). Prediction capability of the uncertainty propagation model and uncertainty results  
After obtaining the high-fidelity responses for all training samples, we constructed a prediction model to compare model parameters and friction coefficients. For the entire model, only the model parameters of the test samples are required as input. We assessed the prediction error of the entire model on 20 test samples. A comparison of the predicted wall friction coefficient distribution with that obtained from the high-fidelity CFD simulation using a randomly selected sample from the test dataset is illustrated in Figure 14. It is evident that the two distributions align well, indicating a strong agreement. Figure 15 presents the prediction errors on all test samples. Notably, the prediction errors are less than 1% across all test samples, indicating the accurate prediction capability of the entire model. This finding demonstrates the utility of the entire model to support uncertainty quantification for large-scale random sampling. Finally, the Latin hypercube method was employed to sample \(10^{6}\) inputs within the nine-dimensional space of uncertain parameters. The samples were processed through the uncertainty propagation model to generate wall friction coefficient distributions. Subsequent statistical evaluation yielded mean values with 99% confidence intervals (Figure 16), revealing substantial uncertainty propagation across the entire airfoil surface.