The black-box nature of deep generative models like Variational Autoencoders (VAEs) limits their practical application in engineering design, particularly in aerospace, where interpretability is crucial for reliability and safety. This work investigates the use of Symbolic Regression (SR) to improve the interpretability of VAE latent spaces for airfoil shape optimization. Two approaches are developed: a latent analysis of a trained β-VAE and a novel SR-VAE model that integrates SR directly into VAE training.

The first approach shows that SR can approximate the decoder via analytical equations linking latent variables to geometric airfoil features. This even enabled parametric reconstruction independent of the decoder, though accuracy was limited for airfoils with extreme thickness or uncommon trailing edges.

The second approach investigates several SR integration strategies, with a per-batch method followed by retraining with fixed equations achieving the best balance between generalizability and reconstruction accuracy. The parametric equations of this final SR-VAE show an improvement over those from the latent analysis while preserving, and in some aspects even improving, the generative capability of the decoder. The latent space itself showed limited change due to the use of warm starts, suggesting that interpretability through SR is improved primarily at the output level.

The practical applicability of the decoders and equations obtained in this work are tested and compared to CST parameterization, using inverse design tests, as well as constrained and unconstrained optimization cases. The SR-VAE decoder consistently showed the highest reconstruction fidelity for inverse design, but limited use in practical optimizations due to its poor generalizability far from the training set mean. Although SR-based parameterizations show limited reconstruction fidelity in inverse design, they demonstrate competitive performance and the fastest convergence in optimization tasks.

Overall, this work demonstrates that SR can bridge the gap between black-box generative models and interpretable, equation-based design, opening new pathways for explainable AI in engineering contexts and beyond.