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In numerous robotics and mechanical engineering applications, among others,data is often constrained on smooth manifolds due to the presence of rotationaldegrees of freedom. Common datadriven and learning-based methods such as neuralordinary differential equations (ODEs), however, typically fail to satisfythese manifold constraints and perform poorly for these applications. Toaddress this shortcoming, in this paper we study a class of neural ordinarydifferential equations that, by design, leave a given manifold invariant, andcharacterize their properties by leveraging the controllability properties ofcontrol affine systems. In particular, using a result due to Agrachev andCaponigro on approximating diffeomorphisms with flows of feedback controlsystems, we show that any map that can be represented as the flow of amanifold-constrained dynamical system can also be approximated using the flowof manifold-constrained neural ODE, whenever a certain controllabilitycondition is satisfied. Additionally, we show that this universal approximationproperty holds when the neural ODE has limited width in each layer, thusleveraging the depth of network instead for approximation. We verify ourtheoretical findings using numerical experiments on PyTorch for the manifoldsS2 and the 3-dimensional orthogonal group SO(3), which are model manifolds formechanical systems such as spacecrafts and satellites. We also compare theperformance of the manifold invariant neural ODE with classical neural ODEsthat ignore the manifold invariant properties and show the superiority of ourapproach in terms of accuracy and sample complexity.

Learning on Manifolds: Universal Approximations Properties using Geometric Controllability Conditions for Neural ODEs