Controllability on landmark manifolds for shapes and neural ODEs

Landmark manifolds consist of a collection of distinct points, and dynamics on this manifold can be used to represent flows, such as solutions of ODEs and flows deforming a shape. We will consider landmark configurations in the Euclidean space and how such configuration can be connected through flows of vector field. For every dimension equal or larger than two, we explicitly describe two vector fields whose flows can connect any pair of landmark configuration regardless of how many points are in the configuration. This property is called the exact universal interpolation property. For the case of dimension one, we show the same result holds for landmark configurations as long as they have the same relative order. In all dimensions, we are able to achieve controllability by combining a constant vector field with a polynomial vector field of degree three.