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1. Characterizing Designs Via Isometric Embeddings: Applications to Airfoil Inverse Design

   https://doi.org/10.1115/1.4063363  

   Chen, Qiuyi; Fuge, Mark (May 2024, Journal of Mechanical 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 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. 

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2. Characterizing Designs via Isometric Embeddings: Applications to Airfoil Inverse Design

   https://doi.org/10.1115/DETC2023-116743  

   Chen, Qiuyi; Fuge, Mark D (August 2023, American Society of Mechanical Engineers)

   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. 

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3. On realization of isometries for higher rank quadratic lattices over number fields

   https://doi.org/10.1090/tran/8670  

   Chan, Wai Kiu; Li, Han (July 2022, Transactions of the American Mathematical Society)

   Let F F be a number field, and n ≥ 3 n\geq 3 be an integer. In this paper we give an effective procedure which (1) determines whether two given quadratic lattices on F n F^n are isometric or not, and (2) produces an invertible linear transformation realizing the isometry provided the two given lattices are isometric. A key ingredient in our approach is a search bound for the equivalence of two given quadratic forms over number fields which we prove using methods from algebraic groups, homogeneous dynamics and spectral theory of automorphic forms. 

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4. On Polish groups admitting non-essentially countable actions

   https://doi.org/10.1017/etds.2020.133  

   KECHRIS, ALEXANDER S.; MALICKI, MACIEJ; PANAGIOTOPOULOS, ARISTOTELIS; ZIELINSKI, JOSEPH (January 2022, Ergodic Theory and Dynamical Systems)

   Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable. 

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5. Fast Diffusion leads to partial mass concentration in Keller–Segel type stationary solutions

   https://doi.org/10.1142/S021820252250018X  

   Carrillo, J. A.; Delgadino, M. G.; Frank, R. L.; Lewin, M. (April 2022, Mathematical Models and Methods in Applied Sciences)

   We show that partial mass concentration can happen for stationary solutions of aggregation–diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions [Formula: see text]. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions [Formula: see text], for homogeneous interaction potentials with higher power. 

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