Continuous-time generative models, such as Flow Matching (FM), construct probability paths to transport between one distribution and another through the simulation-free learning of the neural ordinary differential equations (ODEs). During inference, however, the learned model often requires multiple neural network evaluations to accurately integrate the flow, resulting in a slow sampling speed. We attribute the reason to the inherent (joint) heterogeneity of source and/or target distributions, namely the singularity problem, which poses challenges for training the neural ODEs effectively. To address this issue, we propose a more general framework, termed Switched FM (SFM), that eliminates singularities via switching ODEs, as opposed to using a uniform ODE in FM. Importantly, we theoretically show that FM cannot transport between two simple distributions due to the existence and uniqueness of initial value problems of ODEs, while these limitations can be well tackled by SFM. From an orthogonal perspective, our framework can seamlessly integrate with the existing advanced techniques, such as minibatch optimal transport, to further enhance the straightness of the flow, yielding a more efficient sampling process with reduced costs. We demonstrate the effectiveness of the newly proposed SFM through several numerical examples.

通过使用切换的普通微分方程 (ODEs) 来消除奇点问题，我们提出了一个更通用的框架，Switched FM (SFM)，以解决连续时间生成模型中的采样速度缓慢的问题，并演示了该框架的有效性。

切换流匹配：通过切换ODE来消除奇异性