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Abstract In Topology Optimization (TO) and related engineering applications, physics-constrained simulations are often used to optimize candidate designs given some set of boundary conditions. However, such models are computationally expensive and do not guarantee convergence to a desired result, given the frequent non-convexity of the performance objective. Creating data-based approaches to warm-start these models — or even replace them entirely — has thus been a top priority for researchers in this area of engineering design. In this paper, we present a new dataset of two-dimensional heat sink designs optimized via Multiphysics Topology Optimization (MTO). Further, we propose an augmented Vector-Quantized GAN (VQGAN) that allows for effective MTO data compression within a discrete latent space, known as a codebook, while preserving high reconstruction quality. To concretely assess the benefits of the VQGAN quantization process, we conduct a latent analysis of its codebook as compared to the continuous latent space of a deep AutoEncoder (AE). We find that VQGAN can more effectively learn topological connections despite a high rate of data compression. Finally, we leverage the VQGAN codebook to train a small GPT-2 model, generating thermally performant heat sink designs within a fraction of the time taken by conventional optimization approaches. We show the transformer-based approach is more effective than using a Deep Convolutional GAN (DCGAN) due to its elimination of mode collapse issues, as well as better preservation of topological connections in MTO and similar applications. 

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.