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.

翻译：在众多机器人学和机械工程等应用中，由于旋转自由度的存在，数据通常被约束在光滑流形上。然而，常见的基于数据和学习的方方法（如神经常微分方程）通常会违反这些流形约束，导致在这些应用中性能不佳。为解决这一缺陷，本文研究了一类通过设计保持给定流形不变的神经常微分方程，并利用控制仿射系统的可控性性质来刻画其特性。特别地，基于Agrachev和Caponigro关于用反馈控制系统流逼近微分同胚的结果，我们证明：只要满足特定的可控性条件，任何可表示为流形约束动力系统流的映射，也能用流形约束神经ODE的流来逼近。此外，我们证明当神经ODE每层具有有限宽度时，这种通用逼近性质仍然成立，从而通过增加网络深度来实现逼近。我们利用PyTorch在球面S2和三维正交群SO(3)上进行了数值实验验证理论结果，这两者分别是航天器、卫星等机械系统的模型流形。我们还将流形不变神经ODE与忽略流形不变性质的经典神经ODE进行比较，结果表明我们的方法在精度和样本复杂度方面具有优越性。