We address the problem of learning ordinary differential equations (ODEs) on manifolds. Existing machine learning methods, particularly those using neural networks, often struggle with high computational demands. To overcome this issue, we introduce a geometrically constrained operator-valued kernel that allows us to represent vector fields on tangent bundles of smooth manifolds. The construction of the kernel imposes the geometric constraints that are estimated from the data and ensures the computational feasibility for learning high dimensional systems of ODEs. Once the vector fields are estimated, e.g., by the kernel ridge regression, we need an ODE solver that guarantees the solution to stay on (or close to) the manifold. To overcome this issue, we propose a geometry-preserving ODE solver that approximates the exponential maps corresponding to the ODE solutions. We deduce a theoretical error bound for the proposed solver that guarantees the approximate solutions to lie on the manifold in the limit of large data. We verify the effectiveness of the proposed approach on high-dimensional dynamical systems, including the cavity flow problem, the beating and travelling waves in Kuramoto-Sivashinsky equations, and the reaction-diffusion dynamics.

Deep learning has had remarkable success in robotic perception, but its data-centric nature suffers when it comes to generalizing to ever-changing environments. By contrast, physics-based optimization generalizes better, but it does not perform as well in complicated tasks due to the lack of high-level semantic information and reliance on manual parametric tuning. To take advantage of these two complementary worlds, we present PyPose: a robotics-oriented, PyTorch-based library that combines deep perceptual models with physics-based optimization. PyPose's architecture is tidy and well-organized, it has an imperative style interface and is efficient and user-friendly, making it easy to integrate into real-world robotic applications. Besides, it supports parallel computing of any order gradients of Lie groups and Lie algebras and 2nd-order optimizers, such as trust region methods. Experiments show that PyPose achieves more than 10× speedup in computation compared to the state-of-the-art libraries. To boost future research, we provide concrete examples for several fields of robot learning, including SLAM, planning, control, and inertial navigation.