  Continuous normalizing flows (CNFs) learn an ordinary differential equation
to transform prior samples into data. Flow matching (FM) has recently emerged
as a simulation-free approach for training CNFs by regressing a velocity model
towards the conditional velocity field. However, on constrained domains, the
learned velocity model may lead to undesirable flows that result in highly
unnatural samples, e.g., oversaturated images, due to both flow matching error
and simulation error. To address this, we add a boundary constraint term to
CNFs, which leads to reflected CNFs that keep trajectories within the
constrained domains. We propose reflected flow matching (RFM) to train the
velocity model in reflected CNFs by matching the conditional velocity fields in
a simulation-free manner, similar to the vanilla FM. Moreover, the analytical
form of conditional velocity fields in RFM avoids potentially biased
approximations, making it superior to existing score-based generative models on
constrained domains. We demonstrate that RFM achieves comparable or better
results on standard image benchmarks and produces high-quality
class-conditioned samples under high guidance weight.
