Wasserstein gradient flows (WGFs) describe the evolution of probability distributions in Wasserstein space as steepest descent dynamics for a free energy functional. Computing the full path from an arbitrary initial distribution to equilibrium is challenging, especially in high dimensions. Eulerian methods suffer from the curse of dimensionality, while existing Lagrangian approaches based on particles or generative maps do not naturally improve efficiency through time step tuning. We propose GenWGP, a generative path finding framework for Wasserstein gradient paths. GenWGP learns a generative flow that transports mass from an initial density to an unknown equilibrium distribution by minimizing a path loss that encodes the full trajectory and its terminal equilibrium condition. The loss is derived from a geometric action functional motivated by Dawson Gartner large deviation theory for empirical distributions of interacting diffusion systems. We formulate both a finite horizon action under physical time parametrization and a reparameterization invariant geometric action based on Wasserstein arclength. Using normalizing flows, GenWGP computes a geometric curve toward equilibrium while enforcing approximately constant intrinsic speed between adjacent network layers, so that discretized distributions remain nearly equidistant in the Wasserstein metric along the path. This avoids delicate time stepping constraints and enables stable training that is largely independent of temporal or geometric discretization. Experiments on Fokker Planck and aggregation type problems show that GenWGP matches or exceeds high fidelity reference solutions with only about a dozen discretization points while capturing complex dynamics.

翻译：Wasserstein梯度流（WGFs）描述了概率分布在Wasserstein空间中作为自由能泛函最速下降动力学的演化过程。计算从任意初始分布到平衡态的完整路径极具挑战性，尤其是在高维空间中。欧拉方法存在维度灾难问题，而基于粒子或生成映射的现有拉格朗日方法无法通过时间步长调整自然地提升效率。我们提出GenWGP——一种面向Wasserstein梯度路径的生成式寻径框架。GenWGP通过最小化编码完整轨迹及其终端平衡条件的路径损失函数，学习一个将质量从初始密度输运至未知平衡分布的生成流。该损失函数源于Dawson Gartner大偏差理论中关于相互作用扩散系统经验分布的几何作用泛函。我们同时建立了物理时间参数化下的有限时域作用量，以及基于Wasserstein弧长的重参数化不变几何作用量。通过使用归一化流，GenWGP在保持相邻网络层间近似恒定内禀速度的条件下计算趋近平衡态的几何曲线，使得沿路径离散化的分布在Wasserstein度量下保持近乎等距。这避免了精细的时间步长约束，并实现了与时间或几何离散化几乎无关的稳定训练。在Fokker-Planck和聚集类型问题的实验中，GenWGP仅需约十几个离散点即可匹配甚至超越高保真参考解，同时捕捉复杂动力学行为。