# Machine Learning Prediction of Airfoil Aerodynamic Performance Using Neural Network Ensembles
## 4 Discussion  
This study investigates the influence of random seed initialization and ensemble configuration on the reliability and predictive accuracy of deep learning models for aerodynamic performance, while also introducing a custom-built deep neural network architecture developed and described in detail. A dataset of 112 NACA four-digit airfoils, each defined by 135 coordinate points, was generated to support the training and evaluation process. For each airfoil, aerodynamic data were produced using XFOIL; the input sequence used to run XFOIL is provided in Appendix A.2 for 10 Reynolds numbers ranging from 500,000 to 5,000,000 in 500,000 increments, corresponding to a target speed range of approximately 7 m/s to 78 m/s in standard air conditions. Furthermore, each configuration was evaluated over 20 AoA uniformly distributed between \(-5^{\circ}\) and \(14^{\circ}\). While XFOIL was selected for its computational efficiency, it is known to exhibit reduced accuracy under certain conditions, particularly near stall or in cases of complex boundary layer behavior. As such, the entire dataset inherits the limitations of the underlying simulation tool. Nevertheless, the focus of this present work lies in the machine learning framework, not in the fidelity of the solver itself, and the methodology remains extensible to higher-fidelity datasets such as CFD or experimental measurements in future stages. **The novelty of this work does not lie in the aerodynamic prediction itself, which is a field already well established, but rather, in the in-depth investigation of how random seed initialization impacts model reproducibility and prediction stability**. This study is among the first to share and compare multiple results from the same neural network architecture, showing the range of predictive errors that can emerge solely due to the choice of seed. For identical datasets and code, \(\mathrm{C_{L}}\) prediction errors (MAPE) range from as low as 1.08% (Group B, seed 123) to as high as 2.72% (Group C, ensemble C3). For \(C_{D}\) predictions, the model performance spans an even wider range, with errors from as low as 0.57% (Group B, seed 0) up to 2.62% (Group C, ensemble C1). On average, this indicates a variation of over 250% between the best and worst outcomes, underscoring how heavily reproducibility and accuracy are influenced by seed selection. AoA). A potential limitation of single-seed models could be evident when the dataset is expanded to include more complex geometries, such as NACA five-digit airfoils with more challenging camber lines. In such scenarios, ensembling can help mitigate localized weaknesses by averaging out inconsistent predictions. Thus, ensemble learning introduces a stabilizing effect that enhances generalization in ML models. In regard to tracking the coefficient of determination values applied to the best performing single seed (seed 0) and ensemble (C2), the \(C_{L}\) evaluation is shown in Figure 4, while Figure 5 presents the \(C_{D}\) evaluation. For lift coefficient prediction, both models achieve very high \(R^{2}\) values across all airfoils, but seed 0 shows a slight drop for the last few airfoils while C2 maintains more consistent performance. For drag coefficient predictions, the difference is more visible between the two models tested, as C2 outperforms seed 0 on most airfoils, with higher and smoother \(R^{2}\) values. Overall, applied to all single seeds and ensembles tested in the current work, despite the variability in \(R^{2}\) for \(C_{D}\), all values remain above 0.978, indicating that the model still captures drag trends well. In fact, for the ML architecture tested in this paper, \(R^{2}\) values are consistently above 0.99 for most cases, confirming that the model performs with high reliability, even for the more sensitive drag predictions. Exploring all possible seed values is computationally infeasible, as a 32-bit seed space comprises more than four billion options.