  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.
