Let the Dynamics Flow: Stable Flow Matching Dynamical Systems

Published July 9, 2026 · Category: Robotics

Overview

arXiv:2606.03834v2 Announce Type: replace Abstract: Flow matching has recently emerged as a powerful approach for imitation learning, enabling scalable, expressive, and multimodal motion policies. However, when modeling these policies as dynamical systems, incorporating formal stability guarantees into these generative models is a prerequisite to ensure safe and generalizable robot behaviors, which remains a significant challenge. This paper introduces Stable Flow Matching Dynamical Systems (SFMDS), a novel framework that bridges the gap between highly expressive generative modeling and formal stability guarantees. SFMDS parametrizes dynamical systems via flow matching while constraining the model to satisfy positive invariance and/or Lyapunov stability conditions. We propose two variants: a soft constraint based on a penalty term, and a hard structural constraint embedded directly into the model architecture. We further extend both formulations to Lie groups to robustly handle orientation trajectories. Experiments on benchmark datasets, in simulation, and on a humanoid robot show that SFMDS learns stable, scalable, and multimodal dynamical systems in low- and high-dimensional state spaces, enabling safe and expressive robot motion generation. SFMDS matches or outperforms state-of-the-art methods on unimodal datasets, while substantially improving performance on multimodal datasets, where competing approaches fail to capture multi-modal behaviors. Accompanying source code and video are available at: https://let-the-dynamics-flow.github.io/SFMDS/.

Source

Originally published at arxiv.org.

Related Articles

• Shared Modular Recurrence in Contextual MDPs for Universal Morphology Control
• EmbodiedGen V2: An Agentic, Simulation-Ready 3D World Engine for Embodied AI
• Zero-Human Demonstration End-to-end Autonomous Driving with Trajectory Scorer