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This paper extends the Weak Adversarial Neural Pushforward (WANPF) method to solve the Fokker-Planck equation on Riemannian manifolds. The key insight is leveraging the ambient-space representation of the Laplace-Beltrami operator to evaluate integrals as expectations over samples on the manifold, using globally defined test functions. By constraining a neural pushforward map with manifold retraction, the method enforces probability conservation and manifold membership, enabling mesh-free training with adversarial plane-wave test functions.
Solve Fokker-Planck equations on manifolds without meshes by pushing forward samples with neural networks.
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 $S^2$.
