Equivariant Manifold Neural ODEs and Differential Invariants

In this paper we develop a manifestly geometric framework for equivariant manifold neural ordinary differential equations (NODEs), and use it to analyse their modelling capabilities for symmetric data. First, we consider the action of a Lie group $G$ on a smooth manifold $M$ and establish the equivalence between equivariance of vector fields, symmetries of the corresponding Cauchy problems, and equivariance of the associated NODEs. We also propose a novel formulation of the equivariant NODEs in terms of the differential invariants of the action of $G$ on $M$, based on Lie theory for symmetries of differential equations, which provides an efficient parameterisation of the space of equivariant vector fields in a way that is agnostic to both the manifold $M$ and the symmetry group $G$. Second, we construct augmented manifold NODEs, through embeddings into equivariant flows, and show that they are universal approximators of equivariant diffeomorphisms on any path-connected $M$. Furthermore, we show that the augmented NODEs can be incorporated in the geometric framework and parameterised using higher order differential invariants. Finally, we consider the induced action of $G$ on different fields on $M$ and show how it can be used to generalise previous work, on, e.g., continuous normalizing flows, to equivariant models in any geometry.