Neural Pushforward Samplers for the Fokker-Planck Equation on Embedded Riemannian Manifolds
This provides a mesh-free and chart-free algorithm for computational problems on curved spaces, but it is incremental as it extends an existing method to a new geometric setting.
The paper tackles solving the Fokker-Planck equation on compact embedded Riemannian manifolds by extending a neural pushforward method, demonstrating its ability to capture multimodal invariant distributions on curved spaces like a double-well problem on the two-sphere.
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.