Learning Geometrically-Informed Lyapunov Functions with Deep Diffeomorphic RBF Networks

TL;DR

The paper addresses the challenge of obtaining formal safety guarantees for data-driven dynamical systems by learning Lyapunov-like certificates directly from data. It introduces a diffeomorphic learning framework that encodes the desired Lyapunov geometry in a simple base function $V_b$ and learns a topology-preserving map $oldsymbol{\u0015}$ to form $V_\boldsymbol{\u0015} = V_b \circ \boldsymbol{\u0015}$, ensuring the descent condition on observed trajectories. A novel Deep Diffeomorphic RBF Network (DD-RBFN) realizes the diffeomorphism as a stack of bijective residual RBF layers with Gaussian kernels and weight-box constraints, enabling sup-universal approximation of $\mathcal{C}^2$ diffeomorphisms and a discrete-time flow interpretation. Learning proceeds via MPC-based optimization over layer weights to minimize the empirical Lyapunov risk while enforcing invertibility and Lyapunov constraints, and the approach demonstrates success on LASA shapes, multi-attractor systems, and limit cycles, often outperforming baseline diffeomorphism architectures. This framework offers a principled, geometry-aware pathway to data-driven safety certificates with practical relevance to autonomous systems.

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.

Figures (7)

• Figure 1: Transformations of a 2D space (a) by an invertible (b) and a non-invertible map (c). Invertability requires that the map is unique for each point.
• Figure 2: Depiction of the proposed Deep Diffeomorphic RBFNetwork (DD-RBFN) architecture. In each layer, a residual RBF mapping is applied which induces a state-space transformation $\bm{\phi}(\cdot)$ with a diagonally dominant Jacobian $\bm{J}_\phi(\cdot)$. Through appropriate weight bounds $\rho$, the bijectivity of the mapping is guaranteed.
• Figure 3: Visualization of bound derivation for Gaussian kernel with symmetric covariance. The kernel with diagonal covariance $\bm{D}$ is equivalent to $k$ under a state-space transformation. This transformation is given by the matrix $\bm{Q}$ under which the original $\bm{\Sigma}$ is axis-aligned with the new coordinate axis $\bm{z} = \bm{Q}^\top\bm{x}$.
• Figure 4: Illustration of virtual time-step $\delta$ and resulting implication on the admitted flow-endpoints.
• Figure 5: Successfully learned diffeomorphic Lyapunov functions for $6$ exemplary shapes of the LASA dataset with data trajectories (black) and samples on which the Lyapunov conditions are satisfied (green).

Theorems & Definitions (13)

• Definition 2.1: book:khalil:nonlin
• Theorem 2.2: Lyapunov Stability Theorem, book:khalil:nonlin
• Definition 2.3: book:boumal23
• Proposition 3.1
• proof
• Definition 3.2
• Theorem 4.1: Determinant of Diagonally dominant matrices paper:ostrowski1937
• Theorem 4.2: Bijective Residual RBF Map
• proof
• Proposition 4.3