In recent years, the machine learning community has increasingly embraced the optimal transport (OT) framework for modeling distributional relationships. In this work, we introduce a sample-based neural solver for computing the Wasserstein geodesic between a source and target distribution, along with the associated velocity field. Building on the dynamical formulation of the optimal transport (OT) problem, we recast the constrained optimization as a minimax problem, using deep neural networks to approximate the relevant functions. This approach not only provides the Wasserstein geodesic but also recovers the OT map, enabling direct sampling from the target distribution. By estimating the OT map, we obtain velocity estimates along particle trajectories, which in turn allow us to learn the full velocity field. The framework is flexible and readily extends to general cost functions, including the commonly used quadratic cost. We demonstrate the effectiveness of our method through experiments on both synthetic and real datasets.

翻译：近年来，机器学习领域日益广泛地采用最优输运框架来建模分布关系。本文提出一种基于样本的神经求解器，用于计算源分布与目标分布之间的Wasserstein测地线及其关联速度场。基于最优输运问题的动力学表述，我们将约束优化问题重构为极小极大问题，并利用深度神经网络逼近相关函数。该方法不仅能提供Wasserstein测地线，还能恢复最优输运映射，从而实现对目标分布的直接采样。通过估计最优输运映射，我们获得沿粒子轨迹的速度估计，进而得以学习完整速度场。该框架具有灵活性，可轻松扩展至包括常用二次代价函数在内的一般代价函数。我们通过合成数据集与真实数据集的实验验证了该方法的有效性。