Wasserstein Gradient Flows (WGF) with respect to specific functionals have
been widely used in the machine learning literature. Recently, neural networks
have been adopted to approximate certain intractable parts of the underlying
Wasserstein gradient flow and result in efficient inference procedures. In this
paper, we introduce the Neural Sinkhorn Gradient Flow (NSGF) model, which
parametrizes the time-varying velocity field of the Wasserstein gradient flow
w.r.t. the Sinkhorn divergence to the target distribution starting a given
source distribution. We utilize the velocity field matching training scheme in
NSGF, which only requires samples from the source and target distribution to
compute an empirical velocity field approximation. Our theoretical analyses
show that as the sample size increases to infinity, the mean-field limit of the
empirical approximation converges to the true underlying velocity field. To
further enhance model efficiency on high-dimensional tasks, a two-phase NSGF++
model is devised, which first follows the Sinkhorn flow to approach the image
manifold quickly ($\le 5$ NFEs) and then refines the samples along a simple
straight flow. Numerical experiments with synthetic and real-world benchmark
datasets support our theoretical results and demonstrate the effectiveness of
the proposed methods.

我们提出了神经 Sinkhorn 梯度流（NSGF）模型，该模型使用神经网络来逼近底层 Wasserstein 梯度流的一部分，通过 Sinkhorn 分歧到目标分布的时间变化速度场的参数化，利用速度场匹配训练方案进行样本估计。理论分析表明，随着样本数量无限增加，经验估计的均场近似收敛于真实的底层速度场。为进一步提高模型在高维任务上的效率，我们设计了一个两阶段的 NSGF++ 模型，首先使用 Sinkhorn 流快速接近图像流形（≤5 次 NFE），然后沿着简单的直线流细化样本。通过合成和真实世界基准数据集的数值实验证明了我们的理论结果并证明了所提出方法的有效性。