# Learning Geometrically-Informed Lyapunov Functions with Deep Diffeomorphic RBF Networks

arXiv: https://arxiv.org/abs/2504.02607 · HTML: https://mlsanity.com/paper/2504.02607/

## Key facts

- No citing papers in the map yet (data through 7 Oct 2026).
- Builds on 22 of its 51 references that are in the map.
- First posted on arXiv 3 Apr 2025.

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [A Lyapunov-based Approach to Safe Reinforcement Learning](https://mlsanity.com/paper/1805.07708/) (2018)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems](https://mlsanity.com/paper/1808.00924/) (2018)
- [Data-Efficient Reinforcement Learning with Probabilistic Model Predictive Control](https://mlsanity.com/paper/1706.06491/) (2017)
- [Euclideanizing Flows: Diffeomorphic Reduction for Learning Stable Dynamical Systems](https://mlsanity.com/paper/2005.13143/) (2020)
- [Universal Approximation Property of Neural Ordinary Differential Equations](https://mlsanity.com/paper/2012.02414/) (2020)
- [Universal approximation property of invertible neural networks](https://mlsanity.com/paper/2204.07415/) (2022)
- [Augmenting Reinforcement Learning with Transformer-based Scene Representation Learning for Decision-making of Autonomous Driving](https://mlsanity.com/paper/2208.12263/) (2022)
- [Learning Control Lyapunov Functions from Counterexamples and Demonstrations](https://mlsanity.com/paper/1804.05285/) (2018)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [Diffeomorphically Learning Stable Koopman Operators](https://mlsanity.com/paper/2112.04085/) (2021)
- [Inverse Reinforcement Learning: A Control Lyapunov Approach](https://mlsanity.com/paper/2104.04483/) (2021)
- [Converse Barrier Functions via Lyapunov Functions](https://mlsanity.com/paper/2007.11086/) (2020)
- [Stable Inverse Reinforcement Learning: Policies from Control Lyapunov Landscapes](https://mlsanity.com/paper/2405.08756/) (2024)

## Abstract (arXiv)

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.