CNN-Based Prediction of Aerodynamic Coefficients for Airfoils at Low Reynolds Numbers

Rapid advancements in artificial intelligence (AI), specifically within machine learning and deep learning, have transformed aerodynamic design and analysis by offering new pathways for acceleration. Among these data-driven approaches, Convolutional Neural Networks (CNNs) have demonstrated significant potential in establishing direct mappings from image-based airfoil geometries to aerodynamic coefficients. This capability enables near-instantaneous predictions, which are critical for optimizing applications in both aerospace engineering and renewable energy systems [1, 2].

Conventional evaluation methods such as wind-tunnel experiments and Computational Fluid Dynamics (CFD) remain accurate and widely used; however, they are often resource-intensive, time-consuming, and costly. While high-fidelity solvers (e.g., ANSYS Fluent and OpenFOAM) and lower-cost tools (e.g., XFOIL-based workflows) can provide reliable estimates, they become impractical when thousands of candidate geometries must be screened during iterative early-stage exploration. This motivates surrogate modeling strategies that can reduce turnaround time while maintaining sufficient predictive fidelity for preliminary decision-making [3-5].

Airfoil performance is commonly characterized by the lift coefficient (C_L) and drag coefficient (C_D), which directly influence aerodynamic efficiency, stability, and energy conversion. Recent studies indicate that CNN-based models can approach CFD-comparable accuracy across selected geometries and flow conditions [6, 7]. Nevertheless, key gaps remain: many prior studies rely on relatively small or homogeneous datasets that limit generalization to unseen airfoils and operating conditions [1, 8], and predictive reliability in low Reynolds number regimes critical for small UAVs and small-scale wind turbines remains comparatively underexplored despite the stronger influence of transition and separation phenomena [5]. In addition, maintaining a favorable efficiency–accuracy trade-off as dataset diversity increases calls for systematic dataset construction and validation [6].

To address these gaps, this study develops a CNN framework trained on a diverse dataset of NACA 4-digit airfoils spanning angles of attack from -20° to +20° at Re = 50,000, a representative low-Re regime for small UAVs and micro-turbines. The working hypothesis is that expanding geometric and operating-condition coverage improves out-of-distribution performance while preserving computational efficiency, thereby complementing and partially substituting CFD in early-stage design workflows [9]. The contributions of this work are: (1) construction of an extensive NACA 4-digit dataset with wide angle-of-attack coverage, (2) development of CNN regressors for predicting (C_L) and (C_D) under low-Re conditions, and (3) empirical evaluation demonstrating a practical speed–accuracy trade-off for rapid aerodynamic screening [6, 10-12].

3.1 Result of 25 geometrics

These results confirm that the proposed CNN architecture effectively captures the nonlinear relationship between airfoil geometry and aerodynamic coefficients. Figure 6 presents the training and validation loss curves for predicting the lift (C_L) and drag (C_D) coefficients using 25 airfoil geometries. Both curves exhibit a sharp decline in loss during the first five training epochs, followed by a stable convergence phase, indicating effective model learning and optimization.

Figure 6. Training and validation loss curves for C_L and C_D using 25 geometries

The model achieved high predictive performance, with R² = 0.986 (training) and 0.948 (validation) for C_L, and R² = 0.954 (training) and 0.896 (validation) for C_D. The close agreement between training and validation performance indicates strong generalization capability without signs of significant overfitting.

The training and validation loss curves for both C_L and C_D show a rapid decrease during the initial five epochs, followed by a stable convergence phase. The close proximity between the training and validation losses demonstrates that the CNN effectively learned the aerodynamic relationships while maintaining stable generalization. This behavior indicates that the selected model architecture and hyperparameter configuration are well optimized for the 25 tested airfoil geometries.

Figure 7 illustrates the training and validation R-square curves for predicting the lift (C_L) and drag (C_D) coefficients using 25 airfoil geometries. The R-square curves for both C_L and C_D show a steady increase during the early training epochs and subsequently plateau near 1.0, indicating excellent predictive accuracy and strong model generalization. The close agreement between the training and validation curves confirms that the CNN effectively captured the nonlinear relationship between airfoil geometry and aerodynamic response across all 25 airfoils.

Figure 7. Training and validation R-square curves for C_L and C_D using 25 airfoils

Figure 8 depicts the relationship between the predicted and actual C_L values across the complete dataset. The strong overlap of the two curves over more than 1,400 samples highlight the reliability of the CNN predictions and confirms stable performance across the full range of data.

Figure 8. Graph of the total number of samples

To evaluate robustness under limited data availability, Figure 9 highlights the comparison between actual and predicted C_L values for a reduced subset of 100 representative samples. Despite the smaller sample size, the predicted values maintain close alignment with the reference data, demonstrating that the CNN preserves high predictive accuracy and stable generalization even when evaluated on reduced datasets.

Figure 9. Graph with a total of 100 samples

Drag coefficient prediction performance is summarized in Figure 10, which compares the actual and predicted C_D values for the full dataset. The predicted values consistently follow the reference distribution across the sample range. Quantitatively, this corresponds to a validation R^2 = 0.896, indicating that nearly 90% of the variance in drag coefficient values is captured by the model. The absence of systematic bias confirms that the CNN effectively learns drag-related aerodynamic characteristics.

Figure 10. Comparison of actual vs. predicted C_D

Finally, Figure 11 illustrates the comparison between actual and predicted C_L values across the 25 individual airfoil geometries. The predicted values closely follow the trends of the reference measurements, with only minor deviations observed across the sample index. This result confirms that the CNN maintains consistent predictive accuracy across different airfoil shapes, demonstrating reliable generalization across varying geometric configurations and flow conditions.

Figure 11. Comparison graph between actual and predicted C_L for 25 airfoils

3.2 Result of 33 geometrics

For the dataset comprising 33 airfoil geometries, the CNN model demonstrates consistent and stable performance across both training and validation phases, indicating effective learning as the geometric diversity increases.

Figure 12 illustrates the training and validation loss curves for predicting the lift (C_L) and drag (C_D) coefficients using 33 airfoil geometries. Both loss curves exhibit a rapid decrease during the first five training epochs, followed by stable convergence around epochs 20–30. Quantitatively, the validation loss for C_L converges to approximately 0.017, while the validation loss for C_D stabilizes at approximately 0.0004, indicating effective optimization with minimal overfitting. The small gap between training and validation losses confirms robust generalization despite the increased dataset size.

Figure 12. Training and validation loss for C_L and C_D

To further evaluate regression accuracy, Figure 13 shows the training and validation R-square (R²) curves for C_L and C_D. For both coefficients, the validation R² increases rapidly during the early epochs and stabilizes above 0.93 after approximately 10 epochs. The parallel trends observed between training and validation curves indicate a consistent learning process and demonstrate the CNN’s ability to accurately map geometric features to aerodynamic responses for unseen angles of attack.

Figure 13. Training and validation R² for C_L and C_D

Model consistency is examined through direct comparison between predicted and reference values. Figure 14 compares the actual and predicted C_L values across the full dataset consisting of approximately 1,800 samples. The predicted values closely follow the reference data throughout the sample range, with no visible systematic deviation.

Figure 14. Comparison of actual vs. predicted C_L

Figure 15 highlights the comparison between actual and predicted C_L values for a subset of 100 representative samples. Despite the reduced sample size, the predicted values maintain close alignment with the reference data, indicating that the CNN preserves predictive accuracy and generalization capability when evaluated on limited data.

​Figure 15. Comparison of actual vs. predicted C_L (100 samples)

Drag coefficient prediction performance is summarized in Figure 16, which compares the actual and predicted C_D values across the complete dataset. The predicted results closely follow the reference drag coefficient distribution, and the narrow deviation band reflects stable regression performance.

Figure 16. Comparison of actual vs. predicted C_D

Finally, Figure 17 illustrates the comparison between actual and predicted C_D values for 100 representative samples. The model continues to capture the overall pattern of drag coefficient behavior with minimal variance, confirming that the CNN effectively generalizes aerodynamic features across both full and reduced sample conditions.

Figure 17. Comparison of actual vs. predicted C_D (100 samples)

3.3 Comparison between epoch 30 and epoch 100

The model trained for 30 epochs produced more stable results, characterized by lower validation loss and minimal risk of overfitting. In contrast, extending the training to 100 epochs slightly improved the training R² value to approximately 0.99; however, the validation loss became more fluctuating. This indicates that while longer training enhances fitting accuracy on the training set, it also increases the likelihood of instability and overfitting in the validation phase.

Figure 18 illustrates the convergence behavior of the CNN model for lift coefficient prediction when trained up to 100 epochs. Quantitatively, during the initial training phase (epochs 1–10), the training loss for C_L decreases sharply from approximately 0.09 to 0.01, while the validation loss drops from approximately 0.03 to 0.018, indicating efficient early learning. Between epochs 10 and 40, the training loss further decreases to approximately 0.006, while the validation loss fluctuates mildly around 0.017, suggesting balanced generalization. At epoch 100, the final training and validation losses converge to approximately 0.004 and 0.020, respectively. 

Figure 18. Training loss and validation R-squared curves for C_L and C_D using 33 airfoils at epoch 100

Correspondingly, the training R² increases rapidly from 0.72 to 0.99 within 30 epochs, while the validation R² stabilized around 0.95 with negligible variance thereafter. These results highlight the CNN’s strong capability to capture the nonlinear aerodynamic relationship between airfoil geometry and lift behavior. The plateau of validation performance beyond $\approx$ 40 epochs indicates that the model had reached its optimal convergence point, and further training yielded diminishing improvement in predictive accuracy.

Figure 19 shows the convergence characteristics of the CNN model in predicting the drag coefficient (C_D). During the first ten epochs, the training loss decreases rapidly from approximately 0.009 to 0.002, while the validation loss reduces from approximately 0.007 to 0.0025, indicating fast learning and effective parameter optimization. Beyond epoch 20, both training and validation losses stabilize, with training loss remaining in the range of 0.0008–0.0012 and validation loss fluctuating within 0.001–0.002, demonstrating consistent performance with negligible overfitting. 

Figure 19. Training loss and validation R-square graphs for C_L and C_D using 33 airfoils at epoch 100

In terms of regression accuracy, the training R² for C_D increases from approximately 0.65 in the early epochs to nearly 0.98 by epoch 20, while the validation R² stabilizes in the range of 0.90–0.93. The narrow and stable gap between training and validation curves confirms that the CNN effectively captures drag-related aerodynamic behavior. Importantly, extending training beyond 30–40 epochs does not yield significant improvement in validation accuracy, indicating that additional training cycles provide marginal benefit.

Table 2 shows the comparison of predicted lift coefficient (C_L) values based on dataset size at epoch 30. At epoch 30, validation performance remained acceptable but not yet fully converged for the largest dataset. The validation MSE decreased from 0.0186 (25 NACA) to 0.0173 (33 NACA), then rose slightly to 0.0224 (50 NACA). Similarly, $R_{\text {valid}}^2$ dropped marginally from 0.9485 → 0.9478 → 0.9324, indicating that increasing the dataset size did not automatically enhance generalization when the epoch limit remained fixed. Training metrics were stable (MSE $\approx$ 0.0050–0.0057; $R_{\text {train}}^2 \approx$ 0.984–0.986), but the widening train–validation gap observed for the 50 NACA dataset suggests mild under-training. Overall, the 33 NACA dataset provided the best balance between accuracy and efficiency, while the 50-NACA case would likely benefit from additional training epochs or further hyperparameter tuning to fully exploit the increased data volume.

Table 2. Comparison of predicted values based on data quantity (C_L), epoch 30

As shown in Table 3, extending the training to 100 epochs significantly improves fitting performance for C_L, as reflected by the reduction in training MSE to 0.0025–0.0028 and raising $R_{\text {train}}^2$ to $\approx$ 0.992–0.993. However, validation results were not strictly monotonic: $R_{\text {valid}}^2$ = 0.9611 (25), 0.9418 (33), and 0.9515 (50), with corresponding MSE-valid values of 0.0145, 0.0191, and 0.0163. The best validation performance occurred for 25 NACA, while the 33 NACA case showed a slight overfitting tendency. The 50 NACA dataset recovered some validation strength but still lagged behind the smallest dataset. These patterns imply that while prolonged training enhances fitting, early stopping around 40–60 epochs combined with adaptive learning-rate scheduling may yield superior efficiency and prevent over-optimization.

Table 3. Comparison of predicted values for C_L, epoch 100

For shorter training duration, Table 4 shows that the drag coefficient (C_D) prediction already exhibits high stability at 30 epochs. Validation MSE decreased from 0.0006 → 0.0004 → 0.0004 as the dataset expanded, and $R_{\text {valid}}^2$ improved from 0.8961 (25) to 0.9374 (33), then slightly decreased to 0.9317 (50). Training R² rose steadily (0.9536 → 0.9569 → 0.9697), confirming improved internal representation with larger data. Overall, 33 NACA provided the highest validation accuracy, while 50 NACA performed comparably with a minor gain in training precision. This indicates that for C_D, 30 epochs were already sufficient, with diminishing returns from further data expansion.

Table 4. Comparison of predicted values for C_D, epoch 30

At epoch 100, as summarized in Table 5, the C_D prediction reaches a clear performance plateau. Validation MSE remained within 0.0003–0.0004, and $R_{\text {valid}}^2$ stabilized around 0.93 for all datasets (0.9327 for 25 NACA, 0.9294 for 33 NACA, 0.9332 for 50 NACA). Training R² values remained high ($\approx$ 0.968–0.971), indicating excellent fit without notable overfitting. Compared with epoch 30, the improvement in validation accuracy was negligible, suggesting that additional training cycles yielded no meaningful gain. Consequently, an early-stopping criterion around 30–40 epochs would provide comparable predictive accuracy with reduced computational cost.

Table 5. Comparison of predicted values forC_D, epoch 100

3.4 Summary all test scenarios

Across all test scenarios, the CNN model exhibits consistent convergence behavior, while showing distinct responses to variations in dataset size and training duration for lift (C_L) and drag (C_D) predictions. Table 6 summarizes the comparative trends observed across all experiments and provides a scientific interpretation of the underlying learning behavior. At shorter training durations (30 epochs), the model achieved adequate performance for both coefficients, yet larger datasets (50 NACA) displayed slightly degraded validation accuracy due to underfitting the network did not have sufficient iterations to fully optimize weights across the expanded feature space. Conversely, at longer training (100 epochs), the model achieved near-perfect training R² values ($\approx$ 0.99 for C_L, $\approx$ 0.97 for C_D), but validation improvement plateaued, revealing diminishing returns and minor overfitting tendencies.

Table 6. Comparative trends and scientific interpretation

Scientifically, these phenomena are explained by the bias–variance trade-off and learning saturation effects in deep neural networks. Increasing dataset size without proportional increase in training iterations leads to high bias and incomplete learning of nonlinear aerodynamic features. Meanwhile, excessive training epochs reduce bias but increase variance, causing the model to memorize minor perturbations in training data, particularly in smaller datasets. Furthermore, C_D exhibits smoother and less nonlinear aerodynamic dependency than C_L; therefore, it converges faster and requires fewer epochs to achieve stable generalization.

The observed stabilization of validation loss around epoch 30–40 indicates that the CNN has reached its asymptotic learning plateau, where gradient updates contribute minimally to validation performance. Extending beyond this point mainly refines the training fit but yields negligible gain in predictive generalization.

The findings affirm that CNN training efficiency in aerodynamic prediction depends not only on data volume but also on model data equilibrium: excessive data with limited training induces underfitting, while prolonged training on limited data risks overfitting. The balance point around epoch 30–40 with moderate dataset size ($\approx$ 33 NACA) achieves the optimal bias–variance trade-off, yielding the most stable and generalizable aerodynamic performance predictions.

3.5 Validation against XFOIL and XFLR5 (NACA 0012)

To assess the external validity of the proposed model, the CNN predictions were benchmarked against two widely used low-Reynolds-number aerodynamic solvers, namely XFOIL and XFLR5, using the NACA 0012 airfoil as a reference case. XFOIL employs a panel method coupled with viscous and transition modeling, while XFLR5 is based on lifting-line and panel formulations derived from XFOIL polars, making both suitable baselines for comparison.

Figure 20. Comparison of C_L-C_D polar andC_L-C_D-$\alpha$ curves obtained from CNN predictions and XFOIL/XFLR5 simulations for the NACA 0012 airfoil

Figure 20 illustrates the aerodynamic polar (C_L-C_D) and the lift-to-drag ratio (C_L/C_D) as functions of the angle of attack ($\alpha$) obtained from CNN predictions and FOIL/XFLR5 simulations. The comparison shows that the CNN successfully reproduces the overall polar morphology, including the low-drag bucket and the subsequent rise in C_L/C_D as the angle of attack increases. This agreement indicates that the dominant aerodynamic trends under attached-flow conditions are effectively captured by the data-driven model.

To ensure that the reported accuracy is quantitatively verifiable and self-contained, Table 7 summarizes the predicted C_L/C_D values together with the relative errors of the CNN with respect to XFOIL and XFLR5 over the investigated angle-of-attack range. At a representative operating point of $\alpha$ = 6.75°, the CNN predicts a C_L/C_D value of 24.329, compared with 23.512 obtained from XFOIL and 23.665 from XFLR5. These differences correspond to relative errors of 3.7% and 2.8%, respectively, thereby explicitly substantiating the previously stated “< 4% error” claim through tabulated numerical evidence.

Table 7. Comparison of prediction results from XFOIL, XFLR5, and CNN for NACA 0012

However, Table 7 also reveals that the CNN exhibits notable discrepancies at certain operating conditions. In particular, at an angle of attack of $\alpha$ = 5.75°, the CNN significantly underpredicts the lift-to-drag ratio, yielding a value of 15.71 compared with approximately 25.4 obtained from both XFOIL and XFLR5, corresponding to an error of nearly 38%. This discrepancy is especially critical because this angle lies near the boundary of the low-drag bucket, where small changes in transition or drag level can produce disproportionately large variations in C_L/C_D. Similar deviations are observed at higher angles of attack beyond $\alpha \geqslant$ 7°, where the flow progressively enters transition- and separation-dominated regimes.

From a physical and modeling perspective, these discrepancies arise because the CNN infers aerodynamic behavior solely from airfoil geometry and angle of attack, whereas XFOIL and XFLR5 explicitly account for boundary-layer transition and separation using empirical models. In regions near the onset of the low-drag bucket and close to stall, the aerodynamic response becomes highly nonlinear and sensitive to transition location and separation onset, particularly in drag prediction. As a result, even minor inaccuracies in C_D estimation can lead to large relative errors in the C_L/C_D ratio.

Moreover, the training dataset is typically denser around moderate angles of attack associated with fully attached flow, while near-transition and post-transition conditions are less frequently represented. Consequently, predictions at $\alpha$ = 5.75° and at higher angles rely more on extrapolation than interpolation, which increases uncertainty. This limitation is further amplified by the absence of explicit Reynolds number, Mach number, and transition or tripping parameters in the CNN input space, all of which are implicitly assumed constant but are explicitly modeled in the reference solvers. In addition, the discretized geometric representation employed by the CNN may smooth fine-scale geometric features that strongly influence drag, introducing systematic bias in C_D and, consequently, in C_L/C_D.

Despite these limitations, the close agreement observed in the mid-angle-of-attack range, where the flow remains attached and the aerodynamic response is quasi-linear, demonstrates that the CNN is well suited for rapid aerodynamic screening in early-stage design and optimization. The combined evidence from Figure 20 and Table 7 therefore supports the use of the CNN as a fast surrogate model, while also clearly delineating its predictive boundaries. Future improvements are expected through targeted augmentation of training data in transition and separation regimes, inclusion of additional flow parameters as explicit inputs, and refined regularization strategies to improve robustness across the full aerodynamic envelope.

3.6 Processing time efficiency of CNN, XFLR5, and CFD methods

The computational time analysis highlights the efficiency advantage of the CNN model compared with traditional aerodynamic solvers such as XFLR5 and CFD. In aircraft wing design, balancing accuracy and computational cost remains a persistent challenge, especially for high-fidelity simulations involving complex geometries. The CNN-based prediction framework aims to address this limitation by providing rapid aerodynamic coefficient estimation with acceptable accuracy for design iteration loops.

Based on the benchmark results, the total CNN computation time for preprocessing and simultaneous prediction of C_L and C_D was 32.29 seconds comprising 31.47 seconds for image preprocessing, 0.79 seconds for C_L prediction, and 0.03 seconds for C_D prediction. In contrast, the XFLR5 solver required approximately 82.2 seconds for similar lift and drag computations. The CFD simulation, using a time-step size of 0.001 as reported for the same NACA 0012 profile [23, 24], demanded 16 minutes and 40 seconds to complete a single prediction. These findings demonstrate that CNN offers a speed improvement of roughly 2.5 × over XFLR5 and more than 30 × over CFD, while maintaining comparable accuracy levels.

It is important to note, however, that computational time measurements are influenced by various factors such as hardware capability, solver settings, mesh density, and user expertise. Therefore, while CNN provides a clear acceleration advantage, further benchmarking under standardized conditions is required for fair validation. Previous studies have emphasized the need for explicit benchmarking parameters including hardware specifications, solver tolerances, and dataset consistency to ensure reproducibility and credibility of performance claims [11].

Overall, the results confirm that CNN-based aerodynamic prediction presents a computationally efficient alternative to conventional numerical solvers, capable of drastically reducing turnaround time in the early design phase of aircraft development without compromising prediction reliability.