Learning Dynamics Models with Stable Invariant Sets

Stable invariant sets are an essential notion in the analysis and application of dynamical systems. It is thus of great interest to learn dynamical systems with provable existence of stable invariant sets. However, existing methods can only deal with the stability of discrete equilibria, which hinders many applications. In this paper, we propose a method to ensure that a learned dynamics model has a stable invariant set of general classes. To this end, we modify a base dynamics model using a learnable Lyapunov-like function so that the modified dynamics attain the invariance and the stability of a specific subset. We model such a subset by transforming primitive shapes (e.g., spheres) via a learnable bijective function. We may specify such a primitive shape following prior knowledge of the dynamics if any, or it can also be learned from data. We introduce an example of the implementation of the proposed dynamics models using neural networks and present experimental results that show the validity of the proposed method.