Let the Dynamics Flow: Stable Flow Matching Dynamical Systems

2026-06-02

AI Summary

This work addresses the challenge in imitation learning of simultaneously achieving expressive generative models and provable system stability. The authors propose the SFMDS framework, which integrates flow matching with Lyapunov stability theory to learn high-capacity dynamical systems while incorporating both soft penalty terms and hard structural constraints as stability-inducing inductive biases. The approach is further extended to Lie group manifolds to model high-dimensional state spaces. SFMDS generates multimodal, generalizable robot motion policies with theoretical stability guarantees, demonstrating consistent effectiveness across low- and high-dimensional tasks. Its performance is validated on standard benchmark datasets, in simulation, and through real-world experiments on a humanoid robot.

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.

Problem

• flow matching
• stability guarantees
• imitation learning
• dynamical systems
• robot motion generation

Innovation

• Flow Matching
• Dynamical Systems
• Lyapunov Stability
• Imitation Learning
• Lie Groups