Weak Adversarial Neural Pushforward Method for Fractional Fokker-Planck Equations

TL;DR

This paper introduces a neural network-based method for solving fractional Fokker-Planck equations, leveraging weak formulations and Monte Carlo sampling, enabling efficient high-dimensional anomalous diffusion modeling.

Contribution

The paper extends the Weak Adversarial Neural Pushforward Method to fractional equations, utilizing eigenfunctions of the fractional Laplacian for computational efficiency and mesh-free discretization.

Findings

Validated on seven benchmark problems with alpha=1.5

Achieved close agreement with particle simulations

Method is mesh-free and suitable for high-dimensional problems

Abstract

We extend the Weak Adversarial Neural Pushforward Method (WANPM) to fractional Fokker-Planck equations, in which the classical Laplacian diffusion operator is replaced by the fractional Laplacian of order alpha in (0, 2]. The solution distribution is represented as the pushforward of a simple base distribution through a neural network, and the weak formulation is discretized entirely via Monte Carlo sampling without any temporal mesh. A key computational advantage is that plane-wave test functions are eigenfunctions of the fractional Laplacian, making the operator cost identical to that of classical diffusion for any alpha. We validate the method on seven benchmark problems with alpha = 1.5, spanning one and two spatial dimensions: the steady-state fractional Ornstein--Uhlenbeck (OU) process, a harmonic confining potential, a double-well potential, and a triple-well potential in one dimension, a steady-state 2D double-peak distribution, a time-dependent 2D ring distribution with rotational drift, and a five-dimensional harmonic potential. Each case is benchmarked against particle simulations using symmetric alpha-stable L\'evy increments, and robust statistics confirm close agreement throughout. The method is mesh-free, requires no density evaluation or non-local quadrature, and provides a promising foundation for high-dimensional anomalous diffusion solvers.