Equivariant Manifold Neural ODEs and Differential Invariants

• URL: http://arxiv.org/abs/2401.14131v2
• Date: Thu, 10 Oct 2024 14:22:30 GMT
• Title: Equivariant Manifold Neural ODEs and Differential Invariants
• Authors: Emma Andersdotter, Daniel Persson, Fredrik Ohlsson,
• Abstract summary: We develop a manifestly geometric framework for equivariant manifold neural ordinary differential equations (NODEs) We use it to analyse their modelling capabilities for symmetric data.
• Score: 1.6073704837297416
• License:
• Abstract: 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, based on Lie theory for symmetries of differential equations, of the equivariant manifold NODEs in terms of the differential invariants of the action of $G$ on $M$, 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 flows on the tangent bundle $TM$, and show that they are universal approximators of diffeomorphisms on any connected $M$. Furthermore, we show that universality persists in the equivariant case and that the augmented equivariant manifold NODEs can be incorporated into the geometric framework 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.

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