# Learning Riemannian Stable Dynamical Systems via Diffeomorphisms

arXiv: https://arxiv.org/abs/2211.03169 · HTML: https://mlsanity.com/paper/2211.03169/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 47 references that are in the map.
- First posted on arXiv 6 Nov 2022.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [ImitationFlow: Learning Deep Stable Stochastic Dynamic Systems by Normalizing Flows](https://mlsanity.com/paper/2010.13129/) (2020)
- [Euclideanizing Flows: Diffeomorphic Reduction for Learning Stable Dynamical Systems](https://mlsanity.com/paper/2005.13143/) (2020)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [Learning Riemannian Manifolds for Geodesic Motion Skills](https://mlsanity.com/paper/2106.04315/) (2021)
- [Learning Control Lyapunov Functions from Counterexamples and Demonstrations](https://mlsanity.com/paper/1804.05285/) (2018)
- [Geometric Fabrics: Generalizing Classical Mechanics to Capture the Physics of Behavior](https://mlsanity.com/paper/2109.10443/) (2021)
- [Learning Stable Vector Fields on Lie Groups](https://mlsanity.com/paper/2110.11774/) (2021)
- [Orientation Probabilistic Movement Primitives on Riemannian Manifolds](https://mlsanity.com/paper/2110.15036/) (2021)
- [RMPflow: A Geometric Framework for Generation of Multi-Task Motion Policies](https://mlsanity.com/paper/2007.14256/) (2020)
- [Reactive Motion Generation on Learned Riemannian Manifolds](https://mlsanity.com/paper/2203.07761/) (2022)

## Built on by (most cited first, 17 of 17)

- [Riemannian Flow Matching Policy for Robot Motion Learning](https://mlsanity.com/paper/2403.10672/) (2024)
- [Fast and Robust Visuomotor Riemannian Flow Matching Policy](https://mlsanity.com/paper/2412.10855/) (2024)
- [Neural Contractive Dynamical Systems](https://mlsanity.com/paper/2401.09352/) (2024)
- [Projected Coupled Diffusion for Test-Time Constrained Joint Generation](https://mlsanity.com/paper/2508.10531/) (2025)
- [Unraveling the Single Tangent Space Fallacy: An Analysis and Clarification for Applying Riemannian Geometry in Robot Learning](https://mlsanity.com/paper/2310.07902/) (2023)
- [Contractive Dynamical Imitation Policies for Efficient Out-of-Sample Recovery](https://mlsanity.com/paper/2412.07544/) (2024)
- [PUMA: Deep Metric Imitation Learning for Stable Motion Primitives](https://mlsanity.com/paper/2310.12831/) (2023)
- [Globally Stable Neural Imitation Policies](https://mlsanity.com/paper/2403.04118/) (2024)
- [Fusion Dynamical Systems with Machine Learning in Imitation Learning: A Comprehensive Overview](https://mlsanity.com/paper/2403.19916/) (2024)
- [Stable Autonomous Flow Matching](https://mlsanity.com/paper/2402.05774/) (2024)
- [Geometric Contact Flows: Contactomorphisms for Dynamics and Control](https://mlsanity.com/paper/2506.17868/) (2025)
- [Diffeomorphic Obstacle Avoidance for Contractive Dynamical Systems via Implicit Representations](https://mlsanity.com/paper/2504.18860/) (2025)
- [Contractive Diffusion Policies: Robust Action Diffusion via Contractive Score-Based Sampling with Differential Equations](https://mlsanity.com/paper/2601.01003/) (2026)
- [Learning Geometrically-Informed Lyapunov Functions with Deep Diffeomorphic RBF Networks](https://mlsanity.com/paper/2504.02607/) (2025)
- [Stable Robot Motions on Manifolds: Learning Lyapunov-Constrained Neural Manifold ODEs](https://mlsanity.com/paper/2510.05707/) (2025)
- [A Robust Task-Level Control Architecture for Learned Dynamical Systems](https://mlsanity.com/paper/2511.09790/) (2025)
- [Let the Dynamics Flow: Stable Flow Matching Dynamical Systems](https://mlsanity.com/paper/2606.03834/) (2026)

## Abstract (arXiv)

Dexterous and autonomous robots should be capable of executing elaborated dynamical motions skillfully. Learning techniques may be leveraged to build models of such dynamic skills. To accomplish this, the learning model needs to encode a stable vector field that resembles the desired motion dynamics. This is challenging as the robot state does not evolve on a Euclidean space, and therefore the stability guarantees and vector field encoding need to account for the geometry arising from, for example, the orientation representation. To tackle this problem, we propose learning Riemannian stable dynamical systems (RSDS) from demonstrations, allowing us to account for different geometric constraints resulting from the dynamical system state representation. Our approach provides Lyapunov-stability guarantees on Riemannian manifolds that are enforced on the desired motion dynamics via diffeomorphisms built on neural manifold ODEs. We show that our Riemannian approach makes it possible to learn stable dynamical systems displaying complicated vector fields on both illustrative examples and real-world manipulation tasks, where Euclidean approximations fail.