Stable Autonomous Flow Matching
Christopher Iliffe Sprague, Arne Elofsson, Hossein Azizpour
TL;DR
This work addresses generating samples from physically stable data by strengthening flow-matching generative models with stability guarantees. It introduces autonomous, time-independent marginal CNFs (MCNFs) and CCNFs by augmenting a pseudo-time variable, enabling the application of a stochastic La Salle invariance principle to ensure convergence to and stability on the target data support. The authors define stable MCNF-CCNF families with quadratic potentials and diagonalizable dynamics, derive convergence-rate insights, and establish equivalence between gradients of FM, CFM, and a new Auto-CFM loss. Through moons and circles experiments, Stable-FM demonstrates stable flows where OT-FM diverges, illustrating the practical impact of incorporating control-theoretic stability into flow-based generative modeling. Overall, the paper bridges control theory and generative modeling, offering a principled route to robust sampling for physically stable data and a differential-inclusion perspective on MCNF dynamics.
Abstract
In contexts where data samples represent a physically stable state, it is often assumed that the data points represent the local minima of an energy landscape. In control theory, it is well-known that energy can serve as an effective Lyapunov function. Despite this, connections between control theory and generative models in the literature are sparse, even though there are several machine learning applications with physically stable data points. In this paper, we focus on such data and a recent class of deep generative models called flow matching. We apply tools of stochastic stability for time-independent systems to flow matching models. In doing so, we characterize the space of flow matching models that are amenable to this treatment, as well as draw connections to other control theory principles. We demonstrate our theoretical results on two examples.