Steerable Neural ODEs on Homogeneous Spaces

TL;DR

This paper introduces steerable neural ODEs on homogeneous spaces, extending manifold NODEs to incorporate symmetry groups and equivariant dynamics for vector features.

Contribution

It presents a geometric framework for equivariant neural ODEs on homogeneous spaces, unifying and extending existing models with a focus on symmetry and parallel transport.

Findings

Steerable NODEs are G-equivariant with G-invariant vector fields and connections.

The framework generalizes existing NODE models and flows on Lie groups.

Provides a geometric foundation for continuous-time equivariant feature learning.

Abstract

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.