# ImitationFlow: Learning Deep Stable Stochastic Dynamic Systems by Normalizing Flows

arXiv: https://arxiv.org/abs/2010.13129 · HTML: https://mlsanity.com/paper/2010.13129/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Oct 2020.

## Built on by (most cited first, 30 of 33)

- [SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion](https://mlsanity.com/paper/2209.03855/) (2022)
- [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)
- [ProDMPs: A Unified Perspective on Dynamic and Probabilistic Movement Primitives](https://mlsanity.com/paper/2210.01531/) (2022)
- [Learning Implicit Priors for Motion Optimization](https://mlsanity.com/paper/2204.05369/) (2022)
- [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)
- [Composable Energy Policies for Reactive Motion Generation and Reinforcement Learning](https://mlsanity.com/paper/2105.04962/) (2021)
- [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)
- [Continual Learning from Demonstration of Robotics Skills](https://mlsanity.com/paper/2202.06843/) (2022)
- [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)
- [Scalable and Efficient Continual Learning from Demonstration via a Hypernetwork-generated Stable Dynamics Model](https://mlsanity.com/paper/2311.03600/) (2023)
- [PUMA: Deep Metric Imitation Learning for Stable Motion Primitives](https://mlsanity.com/paper/2310.12831/) (2023)
- [Fusion Dynamical Systems with Machine Learning in Imitation Learning: A Comprehensive Overview](https://mlsanity.com/paper/2403.19916/) (2024)
- [Learning Stabilizable Deep Dynamics Models](https://mlsanity.com/paper/2203.09710/) (2022)
- [Effect of Optimizer, Initializer, and Architecture of Hypernetworks on Continual Learning from Demonstration](https://mlsanity.com/paper/2401.00524/) (2023)
- [Learning Orbitally Stable Systems for Diagrammatically Teaching](https://mlsanity.com/paper/2309.10298/) (2023)
- [Extended Neural Contractive Dynamical Systems: On Multiple Tasks and Riemannian Safety Regions](https://mlsanity.com/paper/2411.11405/) (2024)
- [Data-driven identification of nonlinear dynamical systems with LSTM autoencoders and Normalizing Flows](https://mlsanity.com/paper/2503.03977/) (2025)
- [Diffeomorphic Obstacle Avoidance for Contractive Dynamical Systems via Implicit Representations](https://mlsanity.com/paper/2504.18860/) (2025)
- [Robustly Invertible Nonlinear Dynamics and the BiLipREN: Contracting Neural Models with Contracting Inverses](https://mlsanity.com/paper/2505.03069/) (2025)
- [KoopMotion: Learning Almost Divergence Free Koopman Flow Fields for Motion Planning](https://mlsanity.com/paper/2509.09074/) (2025)
- [Training-free score-based diffusion for parameter-dependent stochastic dynamical systems](https://mlsanity.com/paper/2602.02113/) (2026)
- [Imitation Learning-based Direct Visual Servoing using the Large Projection Formulation](https://mlsanity.com/paper/2406.09120/) (2024)
- [TamedPUMA: safe and stable imitation learning with geometric fabrics](https://mlsanity.com/paper/2503.17432/) (2025)
- [Decision Flow Policy Optimization](https://mlsanity.com/paper/2505.20350/) (2025)
- [A Robust Task-Level Control Architecture for Learned Dynamical Systems](https://mlsanity.com/paper/2511.09790/) (2025)

## Abstract (arXiv)

We introduce ImitationFlow, a novel Deep generative model that allows learning complex globally stable, stochastic, nonlinear dynamics. Our approach extends the Normalizing Flows framework to learn stable Stochastic Differential Equations. We prove the Lyapunov stability for a class of Stochastic Differential Equations and we propose a learning algorithm to learn them from a set of demonstrated trajectories. Our model extends the set of stable dynamical systems that can be represented by state-of-the-art approaches, eliminates the Gaussian assumption on the demonstrations, and outperforms the previous algorithms in terms of representation accuracy. We show the effectiveness of our method with both standard datasets and a real robot experiment.