Equivariant Manifold Neural ODEs and Differential Invariants

Journal article, 2025

In this paper we develop a 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 connected M. Furthermore, we show that the augmented NODEs can be incorporated in the geometric framework and parametrised using higher order differential invariants. Finally, we consider the induced action of G on different fields on M and show how it generalises previous work, e.g., continuous normalizing flows, to equivariant models in any geometry.