用可微分RBF网络学习几何有知的李雅普诺夫函数，实现安全保证
Learning Geometrically-Informed Lyapunov Functions with Deep Diffeomorphic RBF Networks
- 通过可微分RBF网络构建保持拓扑结构的变换，间接逼近目标函数
- 在真实数据上成功学习到满足安全要求的李雅普诺夫函数
- 适合需要形式化安全验证的自主系统研究者
学习型自主系统的实际部署将极大受益于能够从数据中灵活获得安全保证的工具，即证书函数。尽管这类函数的几何特性已清晰理解，但利用机器学习技术合成它们仍具挑战。为此，我们提出一种可微分函数学习框架：将期望输出的先验结构知识编码到一个简单代理函数的几何中，并通过一个表达性强且保持拓扑结构的状态空间变换进行增强。由此实现一个间接函数逼近框架，确保始终位于期望假设空间内。为此，我们引入一种基于RBF网络构造可微分映射的新方法，可在数据附近实现精确、局部的变换。最后，我们在真实世界数据上展示了该方法学习可微分李雅普诺夫函数的能力，并将其应用于多种吸引子系统。
原文摘要 · Abstract (English)
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.
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