Gradient-based regularization for inverse airfoil design

Inverse airfoil design, a cornerstone in aerospace engineering, involves iterative modification of airfoil geometries to achieve specified aerodynamic performance. While recent advances in deep learning have enabled faster alternatives for airfoil shape predictions from pressure coefficient (Cp) distributions, a persistent challenge remains: existing deep-learning-based models often generate airfoils with non-smooth geometries, manifesting as wiggles and kinks on the surface, which are physically nonviable for practical applications. Addressing this critical gap, we introduce a novel gradient-based regularization approach that incorporates first- and second-order spatial gradient terms into the loss function of deep neural networks (DNN), to enforce geometric smoothness while preserving essential aerodynamic characteristics. Results demonstrate that our method achieves a 13% reduction in geometric prediction error with first-order gradient regularization and a 15.5% reduction when both first- and second-order terms are included, compared to baseline deep learning frameworks. Airfoils generated yield pressure coefficient distributions with a 13% lower error relative to vanilla DNN models, underscoring the aerodynamic fidelity of our approach. Furthermore, a spectral analysis of the results show that gradient-regularized models suppress high-frequency oscillations to yield smoother geometries while accurately capturing low-frequency curvature. The improvement is also established against traditional B-spline smoothening and parameterization techniques. Models are tested on unseen angles of attack (AoA), extending the scope of our findings to applications in airfoil morphing. Finally, gradient-regularized DNNs trained across varied Reynold's numbers (Re) and AoA values accurately reconstruct airfoil geometries, demonstrating enhanced generalization over solely inviscid-trained models and applicability of the framework to reliably generate industrially relevant and aerodynamically robust designs.