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.

翻译：我们通过一个适定的常微分方程（ODE）刻画了熵正则化最优输运问题的解。该方法适用于离散边际分布与一般代价函数，除双边际问题外，还可推广至多边际问题及附加线性约束的问题。求解该ODE为最优输运问题提供了一种新的数值方法，其优势在于能同时得到ODE参数（等价于通常的正则化参数）所有中间值对应的解。我们通过若干数值模拟验证了该方法。该ODE的构建还使得我们能够计算ODE参数为$0$（对应仅最小化熵的完全正则化极限问题）时最优代价的导数。