Equivariant Manifold Neural ODEs and Differential Invariants
(2401.14131)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 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.
Overview
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The paper establishes a geometric framework for neural manifold ordinary differential equations (NODEs) that accommodate symmetries using Lie theory.
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It proposes augmented manifold NODEs as universal approximators for equivariant diffeomorphisms on path-connected manifolds, utilizing higher-order differential invariants.
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By extending the capability of NODEs to any symmetry group and manifold, the paper overcomes previous limitations to conservative vector fields.
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The research shows the application potential of NODEs in transforming geometric objects on manifolds, preserving equivariance and having implications in machine learning with symmetry principles.
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The conclusive assessment suggests the geometric framework may lead to more expressive models in various scientific and machine learning domains.
Introduction
The study of neural ordinary differential equations (NODEs) has seen substantial progress, particularly since the proposal of NODEs in their modern form. NODEs originate from the perspective of dynamical systems as the limit of recurrence in infinitely deep networks. Notably, NODEs facilitate the exploration of continuous transformations in deep learning, yielding models with desirable properties, such as reversible transformations that are conceptually attractive for generative models. Recent advancements have extended NODEs to non-Euclidean domains through neural manifold ODEs (manifold NODEs), addressing applications where data exhibit symmetries under a group of transformations and necessitating equivariant NODEs.
Equivariant Manifold NODEs and Differential Invariants
This paper presents a rigorous geometric framework for equivariant manifold NODEs by drawing on classical Lie theory for symmetries in differential equations. This framework allows for the parameterization of equivariant vector fields, regardless of the specific manifold or symmetry group, by using differential invariants of the action of a Lie group on the manifold. In doing so, equivariant differential equations are parameterized. Furthermore, the study introduces augmented manifold NODEs that embody universal approximators for equivariant diffeomorphisms on path-connected manifolds. Higher-order differential invariants permit the construction and parameterization of augmented NODEs within this geometric approach.
Related Work and Contributions
The paper situates itself among seminal works that introduced normalized flows as generative models constructed from simple distributions and NODEs acting on manifolds, and recent efforts that have achieved flows on Riemannian manifolds equivariant under subgroups of isometries. Extending beyond these contributions, the authors establish a comprehensive approach that can accommodate any symmetry group and manifold and address previous restrictions to conservative vector fields by relying on the theory of symmetries in differential equations. The universality of NODEs is a pivotal consideration, with previous studies demonstrating that augmenting NODEs with sufficient dimensions assures their ability to approximate diffeomorphisms. The investigation also appraises the critical task of efficiently parameterizing the space of vector fields on manifolds to articulate expressive models.
Augmentation, Universality, and Applications
The paper explores the construction of universal approximators of equivariant diffeomorphisms through augmented equivariant NODEs on tangent bundles. The differential invariants serve to parameterize these models, offering an insightful geometric understanding of augmentation. Beyond the mathematical underpinning, the authors discuss how NODEs can transform various geometric objects on manifolds, extending from scalar fields to vector fields. The work ensures that transforming a G-equivariant object with a G-equivariant diffeomorphism preserves the equivariance property. This aspect has applications in constructing models for various feature maps in machine learning that uphold symmetry principles, corresponding to 'physically meaningful constraints.
Conclusion
The devised geometric framework for equivariant manifold NODEs enriches the methodological tools available for machine learning applications involving symmetries. By harnessing the foundational principles of differential invariants in equivariant differential equations, the research marks a significant advancement in parameterizing and modeling equivariant flows on manifolds, suitable for a broad spectrum of symmetries and topological intricacies. The implications of such a framework are wide-ranging, from more expressive and accurate models in physical sciences to widespread applications in generative and discriminative machine learning tasks. Future work proposes to further test the capacity of NODEs across varying applications, as well as to explore the relationship between the geometric framework and conservation laws prevalent in many scientific domains.
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