Learning on Manifolds: Universal Approximations Properties using Geometric Controllability Conditions for Neural ODEs
This addresses a critical limitation for applications like spacecraft and satellite modeling where data lies on manifolds, offering a novel method rather than an incremental improvement.
The paper tackles the problem of neural ODEs failing to handle data constrained on smooth manifolds, common in robotics and mechanical engineering, by introducing manifold-invariant neural ODEs that approximate any map representable as a flow of a manifold-constrained dynamical system, achieving superior accuracy and sample complexity in experiments on S2 and SO(3).
In numerous robotics and mechanical engineering applications, among others, data is often constrained on smooth manifolds due to the presence of rotational degrees of freedom. Common datadriven and learning-based methods such as neural ordinary differential equations (ODEs), however, typically fail to satisfy these manifold constraints and perform poorly for these applications. To address this shortcoming, in this paper we study a class of neural ordinary differential equations that, by design, leave a given manifold invariant, and characterize their properties by leveraging the controllability properties of control affine systems. In particular, using a result due to Agrachev and Caponigro on approximating diffeomorphisms with flows of feedback control systems, we show that any map that can be represented as the flow of a manifold-constrained dynamical system can also be approximated using the flow of manifold-constrained neural ODE, whenever a certain controllability condition is satisfied. Additionally, we show that this universal approximation property holds when the neural ODE has limited width in each layer, thus leveraging the depth of network instead for approximation. We verify our theoretical findings using numerical experiments on PyTorch for the manifolds S2 and the 3-dimensional orthogonal group SO(3), which are model manifolds for mechanical systems such as spacecrafts and satellites. We also compare the performance of the manifold invariant neural ODE with classical neural ODEs that ignore the manifold invariant properties and show the superiority of our approach in terms of accuracy and sample complexity.