  We present a flow-based approach to the optimal transport (OT) problem
between two continuous distributions $\pi_0,\pi_1$ on $\mathbb{R}^d$, of
minimizing a transport cost $\mathbb{E}[c(X_1-X_0)]$ in the set of couplings
$(X_0,X_1)$ whose marginal distributions on $X_0,X_1$ equals $\pi_0,\pi_1$,
respectively, where $c$ is a cost function. Our method iteratively constructs a
sequence of neural ordinary differentiable equations (ODE), each learned by
solving a simple unconstrained regression problem, which monotonically reduce
the transport cost while automatically preserving the marginal constraints.
This yields a monotonic interior approach that traverses inside the set of
valid couplings to decrease the transport cost, which distinguishes itself from
most existing approaches that enforce the coupling constraints from the
outside. The main idea of the method draws from rectified flow, a recent
approach that simultaneously decreases the whole family of transport costs
induced by convex functions $c$ (and is hence multi-objective in nature), but
is not tailored to minimize a specific transport cost. Our method is a
single-object variant of rectified flow that guarantees to solve the OT problem
for a fixed, user-specified convex cost function $c$.
