用神经网络求解最优传输路径和速度场，实现高效分布迁移。
Neural solver for Wasserstein Geodesics and optimal transport dynamics
- 将最优传输转为极小极大问题，用神经网络逼近关键函数。
- 不仅能算出源到目标的最优路径，还能还原传输映射与速度场。
- 适合需要精准分布对齐的研究者，如生成模型与动态系统建模。
近年来，机器学习领域越来越重视最优传输（OT）框架在建模分布关系中的应用。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测地线及对应的速场。基于最优传输的动态形式，我们将约束优化问题转化为极小极大问题，并利用深度神经网络近似相关函数。该方法不仅可得到Wasserstein测地线，还能恢复最优传输映射，从而实现从目标分布直接采样。通过估计传输映射，我们获得了粒子轨迹上的速度估计，进而学习完整的速度场。该框架具有灵活性，可扩展至一般代价函数，包括常用的二次代价。我们在合成数据和真实数据集上验证了方法的有效性。
原文摘要 · Abstract (English)
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.
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