保持几何结构的边界流形神经架构
Geometry-Preserving Neural Architectures on Manifolds with Boundary
AI总结：
本文提出统一框架分类几何保持神经架构，针对prox-regular约束集（含边界光滑流形）证明投影神经ODE、中间增强和最终增强架构的逼近定理，实验表明最终增强架构更优。
摘要：
越来越多的神经架构被提出以强制几何约束，包括基于投影的网络、指数映射更新、约束输出层和流形神经ODE。我们通过根据约束在何处以及如何被强制（是在中间层还是仅在最终输出）来组织这些几何保持架构，从而为它们提供一个统一框架。这一视角揭示了现有理论中的若干空白。为了填补这些空白，我们针对prox-regular约束集（包括带边界的光滑流形）证明了投影神经ODE、中间增强架构和最终增强架构的高级逼近定理。在S^2、圆盘、SO(3)上的合成动力学以及SE(3)上的真实蛋白质骨架数据上的数值实验表明，解析更新具有精确可行性，并且最终增强架构具有更简单的结构，在大多数任务中表现更优。当约束集未知时，我们通过小时间热核极限学习投影，表明扩散/流匹配可用作基于数据的投影。此外，我们还展示了强制非凸约束的架构在带边界流形上的路径规划中的有用性。
英文摘要：
A growing number of neural architectures have been proposed to enforce geometric constraints, including projection-based networks, exponential-map updates, constrained output layers, and manifold neural ODEs. We provide a unified framework for these geometry-preserving architectures by organizing them according to where and how constraints are enforced, either throughout the intermediate layers or only at the final output. This perspective reveals several gaps in the existing theory. To address these gaps, we prove high-level approximation theorems for projected neural ODEs, intermediate augmented architectures, and final augmented architectures on prox-regular constraint sets, including smooth manifolds with boundary. Numerical experiments on synthetic dynamics over S^2, the disk, SO(3), together with real-world protein backbone data on SE(3), demonstrate exact feasibility for analytic updates and show that the final augmentation have simpler architecture and outperform in most tasks considered. 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. Moreover, we also the demonstrate the usefulness of the architectures that enforce non-convex constraints for path planning on manifolds with boundary.
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