# Machine Learning Prediction of Airfoil Aerodynamic Performance Using Neural Network Ensembles
## 5 Conclusions  
In addition to absolute accuracy assessed via MAPE and presented in Tables 4, 6 and 8, \(R^{2}\) values were computed for both the best- and worst-performing individual seeds and ensemble configurations, as shown in Table 9. The best performing single seed model (seed 0) achieves the lowest overall MAPE (0.83%) with strong \(R^{2}\) values, summing up an overall 0.9976. In contrast, the worst performing single seed (seed 45) yields a higher overall MAPE of 1.89% and a reduced \(R^{2}\) for drag coefficient prediction (0.9784). Among ensemble models, C2 (seeds 610, 987, 75025) offers the most favorable performance, combining a low overall MAPE of 1.31% with the highest \(R^{2}\) value observed (0.9999 for \(C_{L}\) and 0.9968 for \(C_{D}\)), reflecting better generalization than seed 0. The weakest ensemble, C1 (seeds 17, 89, and 257), exhibits the highest error (2.2% overall MAPE) and comparatively lower reliability in \(R^{2}\) 0.9942. In this study, a total of 91 NACA four-digit airfoils were used for training (details regarding the train/test manual split can be found in Table 2), where for each airfoil, the aerodynamic coefficients were obtained using XFOIL across 10 Reynolds numbers and 20 AoA, resulting in 200 aerodynamic data samples per airfoil. Altogether, 18,299 samples were generated and used as input for training and calibrating the machine learning architecture. In addition to the training set, an independent set of 21 NACA four-digit airfoils, not seen during training, was used exclusively for testing the model performance. Throughout this study, the model was tested with 10 different single seeds and five ensembles in order to investigate the hypothesis that random seed selection plays a significant role in the calibration and predictive behavior of machine learning architectures. Although this study uses a custom-generated dataset of NACA four-digit airfoils with standardized geometry inputs, the model architecture is fully adaptable to benchmark datasets, such as those from CFD repositories; for example, NASA PALMO [30] or RANS alternatives [31]. This adaptability stems from the generic nature of the model inputs, which rely on resampled coordinate data and standard aerodynamic features. While both MAPE and \(R^{2}\) are used to evaluate model performance, MAPE is particularly relevant in the context of aerodynamic prediction, where small numerical deviations \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Category** & **Seed/Ensemble** & **MAPE \(C_{L}\)** & **MAPE \(C_{D}\)** & \(\mathbf{R^{2}}\)\(C_{L}\) & \(\mathbf{R^{2}}\)\(C_{D}\) & **Overall MAPE** & **Overall \(R^{2}\)** \\ \hline Best performing & & & & & & \\ single seed & 1.1\% & 0.57\% & 0.9998 & 0.9954 & 0.83\% & 0.9976 \\ \hline Worst performing & & & & & & \\ single seed & 1.86\% & 1.92\% & 0.9998 & 0.9784 & 1.89\% & 0.9891 \\ \hline Best performing & & & & & & \\ ensemble & C2 (610, 987, 75025) & 1.43\% & 1.19\% & 0.9999 & 0.9968 & 1.31\% & 0.9983 \\ \hline Worst performing & & & & & & \\ ensemble & C1 (17, 89, 257) & 1.78\% & 2.62\% & 0.9998 & 0.9887 & 2.2\% & 0.9942 \\ \hline \end{tabular}
\end{table} 
Table 9: Overall performance of the best/worst tested seeds/ensembles. in lift and drag coefficients can have significant implications. Therefore, for the scope of this study, lower MAPE values reflect higher predictive precision. Nevertheless, all tested models, regardless of seed or ensemble configuration, maintain \(R^{2}\) values exceeding 0.97. This consistently high coefficient of determination demonstrates that the presented architecture is reliable and well-suited to the described dataset, effectively capturing the global variance in the data, which suggests strong generalization across the wide range of aerodynamic conditions considered. The contribution of low MAPE and consistently high \(R^{2}\) indicates both numerical accuracy and strong generalization; therefore, the dataset presented is representative, the features are informative, and the model architecture overall is not overfitting.