Geometry-Preserving Neural Architectures on Manifolds with Boundary
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917244372516864 |
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| author | Elamvazhuthi, Karthik Biswal, Shiba Rosenblum, Kian Katyal, Arushi Qu, Tianli Ma, Grady Sonthalia, Rishi |
| author_facet | Elamvazhuthi, Karthik Biswal, Shiba Rosenblum, Kian Katyal, Arushi Qu, Tianli Ma, Grady Sonthalia, Rishi |
| contents | Preserving geometric structure is important in learning. We propose a unified class of geometry-aware architectures that interleave geometric updates between layers, where both projection layers and intrinsic exponential map updates arise as discretizations of projected dynamical systems on manifolds (with or without boundary). Within this framework, we establish universal approximation results for constrained neural ODEs. We also analyze architectures that enforce geometry only at the output, proving a separate universal approximation property that enables direct comparison to interleaved designs. When the constraint set is unknown, we learn projections via small-time heat-kernel limits, showing diffusion/flow-matching can be used as data-based projections. Experiments on dynamics over S^2 and SO(3), and diffusion on S^{d-1}-valued features demonstrate exact feasibility for analytic updates and strong performance for learned projections |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03082 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometry-Preserving Neural Architectures on Manifolds with Boundary Elamvazhuthi, Karthik Biswal, Shiba Rosenblum, Kian Katyal, Arushi Qu, Tianli Ma, Grady Sonthalia, Rishi Machine Learning Systems and Control Optimization and Control Preserving geometric structure is important in learning. We propose a unified class of geometry-aware architectures that interleave geometric updates between layers, where both projection layers and intrinsic exponential map updates arise as discretizations of projected dynamical systems on manifolds (with or without boundary). Within this framework, we establish universal approximation results for constrained neural ODEs. We also analyze architectures that enforce geometry only at the output, proving a separate universal approximation property that enables direct comparison to interleaved designs. When the constraint set is unknown, we learn projections via small-time heat-kernel limits, showing diffusion/flow-matching can be used as data-based projections. Experiments on dynamics over S^2 and SO(3), and diffusion on S^{d-1}-valued features demonstrate exact feasibility for analytic updates and strong performance for learned projections |
| title | Geometry-Preserving Neural Architectures on Manifolds with Boundary |
| topic | Machine Learning Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2602.03082 |