Equivariant Manifold Neural ODEs and Differential Invariants
Emma Andersdotter Fredrik Ohlsson
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.