# Neural Pushforward Samplers for the Fokker-Planck Equation on Embedded Riemannian Manifolds

arXiv: https://arxiv.org/abs/2603.16239 · HTML: https://mlsanity.com/paper/2603.16239/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 8 references that are in the map.
- First posted on arXiv 17 Mar 2026.

## Builds on (most cited first, 2 of 2)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [A Geometric View of Optimal Transportation and Generative Model](https://mlsanity.com/paper/1710.05488/) (2017)

## Abstract (arXiv)

In this paper, we extend the Weak Adversarial Neural Pushforward Method to the Fokker--Planck equation on compact embedded Riemannian manifolds. The method represents the solution as a probability distribution via a neural pushforward map that is constrained to the manifold by a retraction layer, enforcing manifold membership and probability conservation by construction. Training is guided by a weak adversarial objective using ambient plane-wave test functions, whose intrinsic differential operators are derived in closed form from the geometry of the embedding, yielding a fully mesh-free and chart-free algorithm. Both steady-state and time-dependent formulations are developed, and numerical results on a double-well problem on the two-sphere demonstrate the capability of the method in capturing multimodal invariant distributions on curved spaces.