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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 the 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. 

Many data analysis and design problems involve reasoning about points in high-dimensional space. A common strategy is to embed points from this high-dimensional space into a low-dimensional one. As we will show in this paper, a critical property of good embeddings is that they preserve isometry — i.e., preserving the geodesic distance between points on the original data manifold within their embedded locations in the latent space. However, enforcing isometry is non-trivial for common Neural embedding models, such as autoencoders and generative models. Moreover, while theoretically appealing, it is not clear to what extent enforcing isometry is really necessary for a given design or analysis task. 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 both simple pedagogical examples and for airfoil embeddings using the UIUC airfoil dataset. Our ablation study illustrates that enforcing isometry is necessary to accurately discover latent space clusters — a common analysis method researchers typically perform on low-dimensional embeddings. 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 designer intend to use distance-based analysis measures (such as clustering, k-Nearest Neighbors methods, etc.) 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. 

Design optimization, and particularly adjoint-based multi-physics shape and topology optimization, is time-consuming and often requires expensive iterations to converge to desired designs. In response, researchers have developed Machine Learning (ML) approaches — often referred to as Inverse Design methods — to either replace or accelerate tools like Topology optimization (TO). However, these methods have their own hidden, non-trivial costs including that of data generation, training, and refinement of ML-produced designs. This begs the question: when is it actually worth learning Inverse Design, compared to just optimizing designs without ML assistance? This paper quantitatively addresses this question by comparing the costs and benefits of three different Inverse Design ML model families on a Topology Optimization (TO) task, compared to just running the optimizer by itself. We explore the relationship between the size of training data and the predictive power of each ML model, as well as the computational and training costs of the models and the extent to which they accelerate or hinder TO convergence. The results demonstrate that simpler models, such as K-Nearest Neighbors and Random Forests, are more effective for TO warmstarting with limited training data, while more complex models, such as Deconvolutional Neural Networks, are preferable with more data. We also emphasize the need to balance the benefits of using larger training sets with the costs of data generation when selecting the appropriate ID model. Finally, the paper addresses some challenges that arise when using ML predictions to warmstart optimization, and provides some suggestions for budget and resource management.