# Generative Path-Finding Method for Wasserstein Gradient Flow

arXiv: https://arxiv.org/abs/2604.11519 · HTML: https://mlsanity.com/paper/2604.11519/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 50 references that are in the map.
- First posted on arXiv 13 Apr 2026.

## Builds on (most cited first, 11 of 11)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Adaptive deep density approximation for Fokker-Planck equations](https://mlsanity.com/paper/2103.11181/) (2021)
- [Self-Consistent Velocity Matching of Probability Flows](https://mlsanity.com/paper/2301.13737/) (2023)
- [Score-based Transport Modeling for Mean-Field Fokker-Planck Equations](https://mlsanity.com/paper/2305.03729/) (2023)
- [Deep Variational Free Energy Approach to Dense Hydrogen](https://mlsanity.com/paper/2209.06095/) (2022)
- [Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs](https://mlsanity.com/paper/2303.11205/) (2023)
- [Weak Generative Sampler to Efficiently Sample Invariant Distribution of Stochastic Differential Equation](https://mlsanity.com/paper/2405.19256/) (2024)
- [Computing non-equilibrium trajectories by a deep learning approach](https://mlsanity.com/paper/2210.04042/) (2022)

## Built on by (most cited first, 2 of 2)

- [Wasserstein Residuals: Learning Gradient Flows from Population Dynamics](https://mlsanity.com/paper/2607.04738/) (2026)
- [Simulation-Free Learning of Population Dynamics with Wasserstein Lagrangian Residuals](https://mlsanity.com/paper/2610.03679/) (2026)

## Abstract (arXiv)

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.