Project Optiwing

Enforcing geometric smoothness in deep learning-based inverse airfoil design using spatial gradients.

Inverse airfoil design is a cornerstone in aerospace engineering, involving the iterative modification of airfoil geometries to achieve specified aerodynamic performance. While recent deep learning advances allow faster airfoil shape predictions from pressure coefficient ($C_{p}$) distributions, a persistent challenge remains. Existing deep-learning-based models often generate physically nonviable airfoils with non-smooth geometries, manifesting as wiggles and kinks on the surface.

To address this critical gap, this project (Chivukula et al., 2025) introduces a novel gradient-based regularization approach. By incorporating first- and second-order spatial gradient terms into the loss function of deep neural networks (DNN), we enforce geometric smoothness while preserving essential aerodynamic characteristics.

Left: An illustration of forward vs. inverse airfoil analysis methodologies. Middle: The baseline fully connected DNN architecture predicting the y-coordinates from the pressure distribution. Right: Our proposed training pipeline integrating finite difference schemes to compute gradient-based loss.

Performance and Geometric Accuracy

Instead of relying solely on traditional mean squared error (MSE) which leads to high-variance surfaces, our method supervises the first- and second-order derivatives of the predicted geometry along the chord. This strategy encodes smoothness directly in the output space of the inverse mappin.

The results demonstrated that our approach successfully bridges the gap between data-driven efficiency and physical realism. Our first-order gradient regularized model (G1DNN) achieved a 13% reduction in geometric prediction error, while our model incorporating both first- and second-order terms (G12DNN) achieved a 15.5% reduction compared to baseline deep learning frameworks. Furthermore, airfoils generated by our model yielded pressure coefficient distributions with a 13% lower error relative to vanilla DNN models.

Airfoil shape predictions and MSE distributions for test cases. Notice how the gradient-regularized models (G1DNN and G12DNN) avoid the high-frequency geometric oscillations characteristic of the vanilla DNN, keeping the generated shapes physically viable.

Spectral Analysis and Airfoil Morphing

To quantitatively prove that non-manufacturable wiggles and kinks were suppressed, we performed a dedicated spectral energy analysis of the suction and pressure sides using a one-sided discrete Fourier transform. The spectral analysis confirmed that our gradient-regularized models suppress high-frequency oscillations while accurately capturing low-frequency curvature.

Finally, our practical inverse design surrogate was evaluated across varied operating conditions. The framework was tested on unseen angles of attack (AoA), allowing the model to implicitly project the input pressure distribution onto the learned AoA manifold. This decoupling enables reliable applications in airfoil morphing. Additionally, when trained across varied Reynolds numbers (Re) and AoA values, the network demonstrated enhanced generalization over solely inviscid-trained models, recovering a condition-invariant shape directly from the $C_{p}$ signal.

Left: Spectral energy comparison proving the suppression of high-frequency noise in our G12DNN model versus the ground truth. Right: The model's ability to reliably predict airfoil shapes for morphed target pressure distributions at previously unseen angles of attack.

References

2025

  1. Gradient-based Regularization for Inverse Airfoil Design
    R. Chivukula, S. Pillutla, A. Kalyan, and 4 more authors
    Physics of Fluids, 2025
    Bib