提出可调控的连续动态模型，实现几何对称下的特征演化。
Steerable Neural ODEs on Homogeneous Spaces
- 将特征视为齐性空间上的向量丛截面，通过平行传输建模演化
- 确保在全局对称群作用下保持等变性，支持复杂几何结构学习
- 适用于需要对称性约束的物理模拟与几何建模任务
我们提出了定义在齐性空间 $M=G/H$ 上的可调控神经常微分方程。该模型是流形神经常微分方程（NODEs）的几何推广，能够传输在局部对称群 $H$ 下变换的特征向量。我们将特征解释为 $M$ 上关联向量丛的截面，并将其演化描述为平行传输。这导致了一个耦合的微分方程系统：一个定义在 $M$ 上的流动方程，以及一个作用于特征的调控方程。我们证明，当生成流动的向量场和控制平行传输的联络均为 $G$-不变时，可调控 NODEs 具有 $G$-等变性。此外，我们展示了该框架如何包含现有的节点模型及李群上的连续归一化流。本框架为在齐性空间上学习一般向量值特征的连续时间等变动力学提供了几何基础。
原文摘要 · Abstract (English)
We introduce steerable neural ordinary differential equations on homogeneous spaces $M=G/H$. These models constitute a novel geometric extension of manifold neural ordinary differential equations (NODEs) that transport associated feature vectors transforming under the local symmetry group $H$. We interpret features as sections of associated vector bundles over $M$, and describe their evolution as parallel transport. This results in a coupled system of ODEs consisting of a flow equation on $M$ and a steering equation acting on features. We show that steerable NODEs are $G$-equivariant whenever the vector field generating the flow and the connection governing parallel transport are both $G$-invariant. Furthermore, we demonstrate how steerable NODEs incorporate existing NODE models and continuous normalizing flows on Lie groups. Our framework provides the geometric foundation for learning continuous-time equivariant dynamics of general vector-valued features on homogeneous spaces.
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