Generative Path-Finding Method for Wasserstein Gradient Flow

2026-04-13

AI Summary

This work addresses the challenges of computing paths for high-dimensional Wasserstein gradient flows, which are hindered by the curse of dimensionality and sensitivity to time-step tuning. The authors propose the GenWGP framework, which models the evolution from an initial distribution to an unknown equilibrium via generative flows. By incorporating a geometric action functional grounded in Dawson–Gärtner large deviation theory and employing Wasserstein arc-length parametrization, the method achieves stable, time-step-independent training and approximately equidistant discretization along the path. The approach leverages normalizing flows to optimize a dual-action functional that respects both physical time and geometric invariance, enforcing nearly constant intrinsic velocity between adjacent network layers. In Fokker–Planck and aggregation-type problems, GenWGP attains or surpasses high-fidelity reference solutions using only a dozen discrete points, accurately capturing complex dynamical evolutions.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Natural Language Processing: Generation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

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.

Problem

• Wasserstein gradient flow
• high-dimensional path computation
• curse of dimensionality
• equilibrium distribution
• generative path finding

Innovation

• Wasserstein gradient flow
• generative modeling
• geometric action functional
• normalizing flows
• large deviation theory