# Machine Learning Prediction of Airfoil Aerodynamic Performance Using Neural Network Ensembles
## 3 Results
### Group C--Ensemble Models  
In Group C, the focus is on evaluating the behavior and effectiveness of ensemble learning strategies, unlike Groups A and B, which assess individual seed performance and model sensitivity to random initialization. Group C tests the hypothesis that ensemble models trained on diverse/strategically selected seeds can yield better generalized predictions compared to single-seed models. The ensemble strategy used in this study follows a bagging approach, where multiple models are trained independently. Each model in the ensemble is initialized, similarly to the single seed configuration, with a different random seed. The final prediction is obtained by computing the arithmetic mean of the individual model outputs for each sample. The ensemble models were constructed to explore the influence of seed selection strategies grounded in mathematical fundamentals, specifically:  
* Ensemble C1 was composed of seeds (17, 89, 257), selected as prime numbers, under the hypothesis that their inherent numerical irregularity may induce diverse initialization. * Ensemble C2 includes seeds (610, 987, 75025), which represent Fibonacci numbers--specifically, positions 5, 15, and 25 in the Fibonacci sequence. These seeds were selected to investigate whether growth-based sequence influences model convergence in an ensemble. * Ensemble C3, defined as (\(2^{8}\), \(2^{16}\), \(2^{32}-1\)), is formed by powers of two up to the 32-bit unsigned integer limit. This group was constructed to test edge behavior within the seed space and examine potential instabilities. Based on the rationale for seed selection in Groups A and B and their evaluated performance, the results can be consolidated by separating the tested seeds into two subsets, the three best-performing and the three worst-performing, according to their generalization errors in both \(C_{L}\) and \(C_{D}\); therefore:  
* Ensemble C4 is defined as the ensemble constructed from the best-performing seeds identified in Groups A and B: (0, 123, 999). * Ensemble C5 is formed from the worst-performing seeds, (43, 45, 777), in contrast with C4. The overall model performance in the ensemble setting is presented in Table 7. All ensembles achieve high \(R^{2}\) values (>0.98), confirming robust overall performance. The observed generalization error for \(C_{L}\) in ensemble C3, reaching 2.73%, prompted an investigation into the presence of _possible black swan seeds_. For the purpose of this study, a black swan is defined as any initialization that yields a generalization MAPE for \(C_{L}\) or \(C_{D}\) exceeding 2%, given that the majority of tested seeds, whether standalone (Groups A and B) or ensemble-based (Group C), consistently remained below this threshold. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline \multirow{2}{*}{**Ensembles**} & \multirow{2}{*}{**Training Time**} & **MAPE** & **MAPE** & **R\({}^{2}\)** & **R\({}^{2}\)** \\  & & \(C_{L}\) & \(C_{D}\) & \(C_{L}\) & \(C_{D}\) \\ \hline C1 = (17, 89, 257) & 366.41 s (6.11 min) & 1.78\% & 2.62\% & 0.9998 & 0.9887 \\ C2 = (610, 987, 75025) & 811.98 s (13.53 min) & 1.43\% & 1.19\% & 0.9999 & 0.9968 \\ C3 = \(\left(2^{8},2^{16},2^{32}-1\right)\) & 440.43 s (7.34 min) & 2.72\% & 1.08\% & 0.9998 & 0.997 \\ C4 = (0, 123, 999) & 934.12 s (15.57 min) & 1.82\% & 1.46\% & 0.9999 & 0.9936 \\ C5 = (43, 45, 777) & 1386.45 s (23.11 min) & 1.55\% & 2.01\% & 0.9999 & 0.9893 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Group B—Overall model performance for ensemble setting. Ensemble C3 was constructed using seed values derived from powers of two, extending to the upper limit of a 32-bit unsigned integer: \(2^{8}\), \(2^{16}\), and \(2^{32}-1\). When analyzed individually, the performance of each seed revealed the following:  
* Seed \(2^{8}\) with a 712.02 s (11.87 min) run-time compiles a \(C_{L}\) MAPE of 2.78% and a \(C_{D}\) MAPE of 1.60%, confirming it as a black swan seed.