  Flow-based models are widely used in generative tasks, including normalizing
flow, where a neural network transports from a data distribution $P$ to a
normal distribution. This work develops a flow-based model that transports from
$P$ to an arbitrary $Q$ where both distributions are only accessible via finite
samples. We propose to learn the dynamic optimal transport between $P$ and $Q$
by training a flow neural network. The model is trained to optimally find an
invertible transport map between $P$ and $Q$ by minimizing the transport cost.
The trained optimal transport flow subsequently allows for performing many
downstream tasks, including infinitesimal density ratio estimation (DRE) and
distribution interpolation in the latent space for generative models. The
effectiveness of the proposed model on high-dimensional data is demonstrated by
strong empirical performance on high-dimensional DRE, OT baselines, and
image-to-image translation.
