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

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 $S^2$.

ARDO: A Weak Formulation Deep Neural Network Method for Elliptic and Parabolic PDEs Based on Random Differences of Test Functions

We propose ARDO method for solving PDEs and PDE-related problems with deep learning techniques. This method uses a weak adversarial formulation but transfers the random difference operator onto the test function. The main advantage of this framework is that it is fully derivative-free with respect to the solution neural network. This framework is particularly suitable for Fokker-Planck type second-order elliptic and parabolic PDEs.