Learning Geometrically-Informed Lyapunov Functions with Deep Diffeomorphic RBF Networks

2025-04-03

AI Summary

This work addresses safety verification for learning-based autonomous systems by proposing a data-driven Lyapunov function synthesis method. To overcome the limitation of conventional approaches—namely, their inability to simultaneously guarantee provable stability and geometric consistency—we introduce a novel deep diffeomorphic mapping framework based on radial basis function (RBF) networks. It encodes prior geometric structure into a simple surrogate function and achieves indirect function approximation via topology-preserving state-space transformations, thereby strictly satisfying both stability conditions and physical constraints. Theoretically, our method ensures Lyapunov conditions hold within the hypothesis space, eliminating the need for post-hoc verification. Experiments demonstrate strong generalization and safety guarantees across diverse attractor systems. To our knowledge, this is the first approach to enable end-to-end synthesis of geometrically consistent, formally verifiable stability certificates directly from real-world data.

Technology Category

• Natural Language Processing: Safety and Robustness
• Intelligent Robots: Learning & Optimization for ROB
• Machine Learning: Learning with Manifolds

Application Category

• Security and Privacy: Data transparency and provenance
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

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.

Problem

• Synthesizing geometrically-informed Lyapunov functions for safety guarantees
• Learning diffeomorphic functions with prior structural knowledge
• Constructing precise local transformations using RBF networks

Innovation

• Deep Diffeomorphic RBF Networks for learning
• Topology-preserving state-space transformation framework
• Guaranteed hypothesis space compliance via surrogate