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

2026-03-17

AI Summary

This work proposes a mesh-free neural pushforward method for solving the Fokker–Planck equation on embedded Riemannian manifolds, which rigorously enforces manifold constraints while preserving probability conservation. By introducing adversarial plane-wave test functions in the ambient space and explicitly constructing a weak formulation of the Laplace–Beltrami operator via tangential projection, the approach enables training of the neural pushforward map without relying on automatic differentiation. A manifold retraction mechanism further guarantees that generated samples remain strictly on the manifold. As the first study to extend neural pushforward samplers to Fokker–Planck dynamics on manifolds, the method successfully solves a double-well steady-state problem on the sphere \( S^2 \) and demonstrates its efficacy in both steady-state and time-dependent settings.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Humans and AI: Human-Aware Planning and Behavior Prediction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We extend the Weak Adversarial Neural Pushforward (WANPF) Method to the Fokker--Planck equation posed on a compact, smoothly embedded Riemannian manifold M in $R^n$. The key observation is that the weak formulation of the Fokker--Planck equation, together with the ambient-space representation of the Laplace--Beltrami operator via the tangential projection $P(x)$ and the mean-curvature vector $H(x)$, permits all integrals to be evaluated as expectations over samples lying on M, using test functions defined globally on $R^n$. A neural pushforward map is constrained to map the support of a base distribution into M at all times through a manifold retraction, so that probability conservation and manifold membership are enforced by construction. Adversarial ambient plane-wave test functions are chosen, and their Laplace--Beltrami operators are derived in closed form, enabling autodiff-free, mesh-free training. We present both a steady-state and a time-dependent formulation, derive explicit Laplace--Beltrami formulae for the sphere $S^{n-1}$ and the flat torus $T^n$, and demonstrate the method numerically on a double-well steady-state Fokker--Planck equation on $S^2$.

Problem

• Fokker-Planck equation
• Riemannian manifold
• probability conservation
• manifold constraint
• embedded manifold

Innovation

• Neural Pushforward
• Fokker-Planck Equation
• Riemannian Manifolds
• Laplace–Beltrami Operator
• Mesh-free Training