# Euclideanizing Flows: Diffeomorphic Reduction for Learning Stable Dynamical Systems

arXiv: https://arxiv.org/abs/2005.13143 · HTML: https://mlsanity.com/paper/2005.13143/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2020.

## Built on by (most cited first, 30 of 32)

- [Riemannian Flow Matching Policy for Robot Motion Learning](https://mlsanity.com/paper/2403.10672/) (2024)
- [Deep Generative Models in Robotics: A Survey on Learning from Multimodal Demonstrations](https://mlsanity.com/paper/2408.04380/) (2024)
- [Fast and Robust Visuomotor Riemannian Flow Matching Policy](https://mlsanity.com/paper/2412.10855/) (2024)
- [Learning Riemannian Stable Dynamical Systems via Diffeomorphisms](https://mlsanity.com/paper/2211.03169/) (2022)
- [Neural Contractive Dynamical Systems](https://mlsanity.com/paper/2401.09352/) (2024)
- [Learning Stable Vector Fields on Lie Groups](https://mlsanity.com/paper/2110.11774/) (2021)
- [Robot Skill Adaptation via Soft Actor-Critic Gaussian Mixture Models](https://mlsanity.com/paper/2111.13129/) (2021)
- [Symplectic Adjoint Method for Exact Gradient of Neural ODE with Minimal Memory](https://mlsanity.com/paper/2102.09750/) (2021)
- [Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees](https://mlsanity.com/paper/2402.08090/) (2024)
- [RMP2: A Structured Composable Policy Class for Robot Learning](https://mlsanity.com/paper/2103.05922/) (2021)
- [Projected Coupled Diffusion for Test-Time Constrained Joint Generation](https://mlsanity.com/paper/2508.10531/) (2025)
- [Reactive Motion Generation on Learned Riemannian Manifolds](https://mlsanity.com/paper/2203.07761/) (2022)
- [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)
- [Elastic Motion Policy: An Adaptive Dynamical System for Robust and Efficient One-Shot Imitation Learning](https://mlsanity.com/paper/2503.08029/) (2025)
- [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)
- [KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-Zero Training Loss](https://mlsanity.com/paper/2102.11923/) (2021)
- [Learning Dynamical Systems Encoding Non-Linearity within Space Curvature](https://mlsanity.com/paper/2403.11948/) (2024)
- [Geometric Contact Flows: Contactomorphisms for Dynamics and Control](https://mlsanity.com/paper/2506.17868/) (2025)
- [Linearization and Identification of Multiple-Attractor Dynamical Systems through Laplacian Eigenmaps](https://mlsanity.com/paper/2202.09171/) (2022)
- [Learning Orbitally Stable Systems for Diagrammatically Teaching](https://mlsanity.com/paper/2309.10298/) (2023)
- [Single-Shot Learning of Stable Dynamical Systems for Long-Horizon Manipulation Tasks](https://mlsanity.com/paper/2410.01033/) (2024)
- [Extended Neural Contractive Dynamical Systems: On Multiple Tasks and Riemannian Safety Regions](https://mlsanity.com/paper/2411.11405/) (2024)
- [Diffeomorphic Obstacle Avoidance for Contractive Dynamical Systems via Implicit Representations](https://mlsanity.com/paper/2504.18860/) (2025)
- [KoopMotion: Learning Almost Divergence Free Koopman Flow Fields for Motion Planning](https://mlsanity.com/paper/2509.09074/) (2025)
- [Learning to Represent Surroundings, Anticipate Motion and Take Informed Actions in Unstructured Environments](https://mlsanity.com/paper/2407.10383/) (2024)
- [TamedPUMA: safe and stable imitation learning with geometric fabrics](https://mlsanity.com/paper/2503.17432/) (2025)
- [Learning Geometrically-Informed Lyapunov Functions with Deep Diffeomorphic RBF Networks](https://mlsanity.com/paper/2504.02607/) (2025)

## Abstract (arXiv)

Robotic tasks often require motions with complex geometric structures. We present an approach to learn such motions from a limited number of human demonstrations by exploiting the regularity properties of human motions e.g. stability, smoothness, and boundedness. The complex motions are encoded as rollouts of a stable dynamical system, which, under a change of coordinates defined by a diffeomorphism, is equivalent to a simple, hand-specified dynamical system. As an immediate result of using diffeomorphisms, the stability property of the hand-specified dynamical system directly carry over to the learned dynamical system. Inspired by recent works in density estimation, we propose to represent the diffeomorphism as a composition of simple parameterized diffeomorphisms. Additional structure is imposed to provide guarantees on the smoothness of the generated motions. The efficacy of this approach is demonstrated through validation on an established benchmark as well demonstrations collected on a real-world robotic system.