Let the Dynamics Flow: Stable Flow Matching Dynamical Systems

• URL: http://arxiv.org/abs/2606.03834v1
• Date: Tue, 02 Jun 2026 16:17:26 GMT
• Title: Let the Dynamics Flow: Stable Flow Matching Dynamical Systems
• Abstract summary: This paper introduces Stable Flow Matching Dynamical Systems (SFMDS), a novel framework that bridges the gap between high-capacity generative modeling and formal Lyapunov stability guarantees.<n>SFMDS parametrizes dynamical systems via flow matching while simultaneously constraining the model to a family of stable solutions.<n> 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.
• Score: 13.796642439884849
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Flow matching has recently emerged as a powerful approach for imitation learning, enabling scalable, expressive, and multimodal motion policies. However, incorporating formal stability guarantees into these generative models, a prerequisite to ensure safe and generalizable robot behaviors, remains a significant challenge. While modeling robot motions as dynamical systems allows for such stability-based inductive biases, existing frameworks struggle to capture the rich action distributions inherent in complex robotic tasks. This paper introduces Stable Flow Matching Dynamical Systems (SFMDS), a novel framework that bridges the gap between high-capacity generative modeling and formal Lyapunov stability guarantees. SFMDS parametrizes dynamical systems via flow matching while simultaneously constraining the model to a family of stable solutions. We propose two variants: a soft constraint based on a penalty term, and a hard structural constraint embedded directly in the model architecture. We further extend both formulations to Lie groups. 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.

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