Learning Geometrically-Informed Lyapunov Functions with Deep Diffeomorphic RBF Networks
This work addresses the problem of obtaining safety guarantees for learning-based autonomous systems, which is crucial for their practical deployment, but it appears incremental as it builds on existing geometrical understandings and machine learning techniques.
The paper tackles the challenge of synthesizing safety certificate functions like Lyapunov functions from data by proposing a diffeomorphic function learning framework that encodes prior structural knowledge into a surrogate function and uses topology-preserving transformations to ensure the output remains in the desired hypothesis space. The result is demonstrated by learning diffeomorphic Lyapunov functions from real-world data and applying the method to different attractor systems, though no concrete performance numbers are provided.
The practical deployment of learning-based autonomous systems would greatly benefit from tools that flexibly obtain safety guarantees in the form of certificate functions from data. While the geometrical properties of such certificate functions are well understood, synthesizing them using machine learning techniques still remains a challenge. To mitigate this issue, we propose a diffeomorphic function learning framework where prior structural knowledge of the desired output is encoded in the geometry of a simple surrogate function, which is subsequently augmented through an expressive, topology-preserving state-space transformation. Thereby, we achieve an indirect function approximation framework that is guaranteed to remain in the desired hypothesis space. To this end, we introduce a novel approach to construct diffeomorphic maps based on RBF networks, which facilitate precise, local transformations around data. Finally, we demonstrate our approach by learning diffeomorphic Lyapunov functions from real-world data and apply our method to different attractor systems.