Geometry-Preserving Neural Architectures on Manifolds with Boundary

• URL: http://arxiv.org/abs/2602.03082v1
• Date: Tue, 03 Feb 2026 04:09:39 GMT
• Title: Geometry-Preserving Neural Architectures on Manifolds with Boundary
• Authors: Karthik Elamvazhuthi, Shiba Biswal, Kian Rosenblum, Arushi Katyal, Tianli Qu, Grady Ma, Rishi Sonthalia,
• Abstract summary: We propose a class of geometry-aware architectures that interleave geometric updates between layers.<n>We establish universal approximation results for constrained neural ODEs.<n>Experiments on dynamics over S2 and SO(3), and diffusion on Sd-1-valued features demonstrate exact feasibility for analytic updates.
• Score: 3.6352834408416412
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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

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