  Continuous normalizing flows (CNFs) learn the probability path between a
reference and a target density by modeling the vector field generating said
path using neural networks. Recently, Lipman et al. (2022) introduced a simple
and inexpensive method for training CNFs in generative modeling, termed flow
matching (FM). In this paper, we re-purpose this method for probabilistic
inference by incorporating Markovian sampling methods in evaluating the FM
objective and using the learned probability path to improve Monte Carlo
sampling. We propose a sequential method, which uses samples from a Markov
chain to fix the probability path defining the FM objective. We augment this
scheme with an adaptive tempering mechanism that allows the discovery of
multiple modes in the target. Under mild assumptions, we establish convergence
to a local optimum of the FM objective, discuss improvements in the convergence
rate, and illustrate our methods on synthetic and real-world examples.
