math.NAMar 17, 2026

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

Authors: Andrew Qing He, Wei Cai

Organizations: Department of Mathematics, Southern Methodist University, Dallas, TX 75275.

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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.

Figures & tables

ParameterSymbolValue
Diffusion coefficientσ\sigma0.50.5
Base dimensiondd33
Generator hidden width—3232
Generator depth—22 layers
Number of test functionsKK200200
Monte Carlo batch sizeMsM_{s}200200
Table 1: Hyperparameters for the S2S^{2} double-well experiment.
Figure 1: Left: the double-well potential V(x,y,z)=α(x2−1)2+βz2V(x,y,z)=\alpha(x^{2}-1)^{2}+\beta z^{2} on S2S^{2} , shown in longitude–latitude projection. The two potential minima ( V=0V=0 ) are located at longitude 0∘0^{\circ} and ±180∘\pm 180^{\circ} on the equator, corresponding to (±1,0,0)(\pm 1,0,0) . Right: density histogram of 8,0008{,}000 samples generated by the trained neural pushforward map in the same coordinate system. Mass concentrates near both well bottoms and is absent near the poles, consistent with the Gibbs distribution ρ∞∝e−2V/σ2\rho_{\infty}\propto e^{-2V/\sigma^{2}} for σ=0.5\sigma=0.5 .
Figure 2: Training and sampling diagnostics for the S2S^{2} double-well experiment. Left: adversarial loss versus training step (log scale). The loss rises during early generator exploration and then decreases as training converges to a stationary min-max point near step 2,0002{,}000 . Center: 8,0008{,}000 generated samples on S2S^{2} , coloured by their xx -coordinate value (red near x=+1x=+1 , blue near x=−1x=-1 ); gold stars mark the two well bottoms (±1,0,0)(\pm 1,0,0) . The generator correctly concentrates mass around both minima symmetrically. Right: Mollweide equal-area projection of the same samples; dashed gold lines indicate the longitudes of the well bottoms. Samples cluster near 0∘0^{\circ} and ±180∘\pm 180^{\circ} longitude, confirming two-well capture, with near-uniform spreading in longitude about each minimum and strong equatorial confinement consistent with the βz2\beta z^{2} term.
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