  We characterize the solution to the entropically regularized optimal
transport problem by a well-posed ordinary differential equation (ODE). Our
approach works for discrete marginals and general cost functions, and in
addition to two marginal problems, applies to multi-marginal problems and those
with additional linear constraints. Solving the ODE gives a new numerical
method to solve the optimal transport problem, which has the advantage of
yielding the solution for all intermediate values of the ODE parameter (which
is equivalent to the usual regularization parameter). We illustrate this method
with several numerical simulations. The formulation of the ODE also allows one
to compute derivatives of the optimal cost when the ODE parameter is $0$,
corresponding to the fully regularized limit problem in which only the entropy
is minimized.
