In this paper we study continuum-marginal optimal transport. Given a time-continuous family of probability marginals, the problem is to recover the minimum-energy velocity field whose flow reproduces every marginal. This problem is the continuum limit of the classical two-marginal Benamou--Brenier formulation, and also the deterministic limit of the Nelson problem of stochastic optimal transport. We propose a practical mesh-free solver for this problem. The weak continuity equation is embedded in a reproducing kernel Hilbert space, yielding a sample-only objective that requires no spatial discretization. The velocity is parametrized by any linear-in-parameters dictionary or neural network, and is optimized by mini-batch stochastic methods. Synthetic experiments confirm that the method achieves accurate drift recovery and marginal consistency. The same computational framework also applies to the stochastic Nelson problem.

翻译：本文研究了连续边际最优传输问题。给定一个时间连续的概率边际族，该问题旨在恢复最小能量速度场，其流场可重现每个边际分布。该问题是经典双边际Benamou-Brenier公式的连续极限，亦是纳尔逊随机最优传输问题的确定性极限。我们提出了一种实用的无网格求解器。弱连续性方程被嵌入再生核希尔伯特空间，形成仅需样本数据的优化目标，无需空间离散化。速度场通过任意线性参数化字典或神经网络进行参数化，并采用小批量随机方法进行优化。合成实验证实该方法可实现精确的漂移恢复与边际一致性。相同的计算框架同样适用于纳尔逊随机最优传输问题。