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.

翻译：在数据样本代表物理稳定状态的场景中，通常假设数据点对应于能量景观的局部极小值。在控制理论中，能量可作为有效的李雅普诺夫函数这一事实已得到公认。然而，尽管存在多个具有物理稳定数据点的机器学习应用，但文献中控制理论与生成模型之间的联系仍较为稀疏。本文聚焦于此类数据及名为流匹配的新兴深度生成模型类别，将时不变系统的随机稳定性工具应用于流匹配模型。通过这一方法，我们刻画了适用于该处理的流匹配模型空间，并建立了与其他控制理论原则的联系。本文通过两个实例验证了理论成果。