Physics-informed geometric operators in generative airfoil design with hybrid VAEs

Abstract This work investigates the use of physics-informed geometric operators in tandem with latent variable models to support surrogate learning, dimensionality reduction, and generative design of airfoils. A baseline dataset was constructed via a NURBS-based airfoil parametric model with physically interpretable design variables, and discretized using two schemes: a uniform parametric and a uniform arc-length sampling. A hybrid Variational AutoEncoder (VAE) with the addition of convolutional layers was developed and iteratively refined to evade shape invalidities. A systematic comparison of reconstruction accuracy, robustness, and diversity showed that a loss function based on the mean sum of squared distances had the best performance with sufficient stability during model optimization. However, this can be only established when the reconstruction and Kullback–Leibler terms in the $\beta$-VAE objective function are weighted via an appropriately selected $\beta$ value. Additionally, augmenting geometry with physics-correlated high-level descriptors, such as geometric moments, further improves latent-space quality. Among the tested operators, third-order geometric moments yielded the most consistent robustness gains. Discretization and achieved diversity proved to be linked, with uniform arc-length spacing achieving the best reconstruction accuracy, but with many near-identical designs that degraded the resulting diversity. In contrast, uniform parametric spacing exhibited higher diversity without the need for any special treatment of design distributions and diversity quantification measures. This study consolidates practical guidelines on architecture, loss-function scaling, physics-informed features, and quantification protocols for reliable, data-efficient airfoil generative design with VAEs.