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

• URL: http://arxiv.org/abs/2603.16239v2
• Date: Wed, 18 Mar 2026 01:00:59 GMT
• Title: Neural Pushforward Samplers for the Fokker-Planck Equation on Embedded Riemannian Manifolds
• Abstract summary: We extend the Weak Adversarial Neural Pushforward Method to the Fokker--Planck equation.<n>Training is guided by a weak adversarial objective using ambient plane-wave test functions.<n> numerical results on a double-well problem on the two-sphere demonstrate the capability of the method in capturing multimodal invariant distributions.
• Score: 2.9762212209762606
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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