Geometry-Preserving Neural Architectures on Manifolds with Boundary

2026-02-03

AI Summary

This work addresses the challenge of preserving geometric constraints when training neural networks on manifolds with boundary. We propose a class of geometry-aware neural architectures that interpret network layers as discretizations of projected dynamical systems on manifolds, explicitly enforcing manifold constraints through geometric update operations such as exponential maps and projections. A universal approximation theorem is established for constrained neural ODEs, distinguishing between output-level and layer-wise constraint mechanisms. Furthermore, we introduce a data-driven projection method grounded in the heat kernel limit. Evaluated on tasks including dynamics modeling on $S^2$ and $\mathrm{SO}(3)$, as well as feature diffusion on $S^{d-1}$, our approach achieves high-precision geometric preservation—using either analytical or learned projections—and significantly outperforms existing methods.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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

Problem

• manifolds with boundary
• geometry-preserving
• neural architectures
• geometric constraints
• constrained learning

Innovation

• geometry-preserving neural networks
• manifolds with boundary
• neural ODEs
• universal approximation
• heat-kernel projections