Weak Adversarial Neural Pushforward Method for Fractional Fokker-Planck Equations

Abstract

We extend the Weak Adversarial Neural Pushforward Method (WANPM) to fractional Fokker-Planck equations (fFPE), in which the classical Laplacian diffusion operator is replaced by the fractional Laplacian $(-Δ)^{α/2}$ for $α\in (0, 2]$. The solution distribution is represented not as an explicit probability density function but as the pushforward of a simple base distribution through a time-parameterized neural network $F_\vartheta(t, x_0, r)$, which enforces the initial condition exactly by construction. The weak formulation of the fFPE is discretized via Monte Carlo sampling entirely without temporal discretization, and the resulting min-max objective is optimized adversarially against a set of plane-wave test functions. A key computational advantage is that plane waves are eigenfunctions of the fractional Laplacian, so $(-Δ_x)^{α/2} f = |w|^αf$ is computed exactly and at no additional cost for any $α$. We validate the method on a one-dimensional fractional Fokker-Planck equation with a quadratic confining potential and $α= 1.5$, using a particle simulation based on symmetric $α$-stable Levy increments as a benchmark. The learned solution faithfully reproduces the transient probability distribution over $t \in [0, 2]$, and robust statistics confirm close agreement with the particle simulation, while standard deviation comparisons highlight why second-moment metrics are inappropriate for heavy-tailed ($α< 2$) distributions.

Figures (4)

• Figure 1: Quadratic confining potential $V(x) = \tfrac{1}{2}kx^2$ (red, left axis) and initial condition $\rho_0 = \mathcal{N}(1.0, 0.09)$ (blue, right axis) for the fractional Fokker-Planck experiment with $\alpha = 1.5$, $k = 1.0$. The initial mass is offset from the potential minimum at $x = 0$; the dynamics transport it toward the origin while fractional diffusion develops heavy tails.
• Figure 2: Training history over 1,000 epochs for the fractional Fokker-Planck problem with $\alpha = 1.5$. Left: training loss \ref{['eq:total_loss']} on a log scale. Right: $\ell^2$ norm of the weak-form residual vector $(R^{(1)}, \ldots, R^{(K)})$ on a log scale. Both curves exhibit the rapid initial convergence and adversarial oscillation typical of min-max training, with the loss stabilizing around $10^{-2}$ and the residual norm around $10^{1}$ (note that the residual norm is summed over $K = 2{,}000$ test functions).
• Figure 3: Fractional Fokker-Planck evolution ($\alpha = 1.5$): neural network samples (blue) versus particle simulation (green) at eight time points. Both histograms are density-normalized over $[-3, 3]$ with 100 bins. The neural network faithfully tracks the drift of the bulk toward the potential minimum at $x = 0$ and the development of algebraically heavy tails over time.
• Figure 4: Robust comparison metrics over 50 time snapshots for the fractional Fokker-Planck problem ($\alpha = 1.5$): mean (upper left), median (upper right), IQR (center left), MAD (center right), percentile bands (lower left), and standard deviation (lower right). The neural network (solid blue) closely tracks the particle simulation (dashed green) for all robust metrics. The standard deviation panel demonstrates the well-known instability of second-moment statistics for $\alpha < 2$ Lévy processes: the particle simulation exhibits large jumps due to occasional extreme samples, while the neural network underestimates the tails. Standard deviation should not be used as a comparison metric for $\alpha < 2$.

Theorems & Definitions (4)

• Remark 1: Comparison with the fractional diffusion normalization
• Remark 2: Structure of the weak form
• Remark 3: Overparameterized base distributions
• Remark 4: Variance and heavy tails