Characterizing Designs via Isometric Embeddings: Applications to Airfoil Inverse Design

Many design problems involve reasoning about points in high-dimensional space. A common strategy is to first embed these high-dimensional points into a low-dimensional latent space. We propose that a good embedding should be isometric---i.e., preserving the geodesic distance between points on the data manifold in the latent space. However, enforcing isometry is non-trivial for common Neural embedding models such as autoencoders. Moreover, while theoretically appealing, it is unclear to what extent is enforcing isometry necessary for a given design analysis. This paper answers these questions by constructing an isometric embedding via an isometric autoencoder, which we employ to analyze an inverse airfoil design problem. Specifically, the paper describes how to train an isometric autoencoder and demonstrates its usefulness compared to non-isometric autoencoders on the UIUC airfoil dataset. Our ablation study illustrates that enforcing isometry is necessary for accurately discovering clusters through the latent space. We also show how isometric autoencoders can uncover pathologies in typical gradient-based Shape Optimization solvers through an analysis on the SU2-optimized airfoil dataset, wherein we find an over-reliance of the gradient solver on angle of attack. Overall, this paper motivates the use of isometry constraints in Neural embedding models, particularly in cases where researchers or designers intend to use distance-based analysis measures to analyze designs within the latent space. While this work focuses on airfoil design as an illustrative example, it applies to any domain where analyzing isometric design or data embeddings would be useful.