Paper
Weak Adversarial Neural Pushforward Method for Fractional Fokker-Planck Equations
Authors
Andrew Qing He, Wei Cai
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.
Metadata
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.12869v1</id>\n <title>Weak Adversarial Neural Pushforward Method for Fractional Fokker-Planck Equations</title>\n <updated>2026-03-13T10:15:54Z</updated>\n <link href='https://arxiv.org/abs/2603.12869v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.12869v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.NA'/>\n <category scheme='http://arxiv.org/schemas/atom' term='math.AP'/>\n <published>2026-03-13T10:15:54Z</published>\n <arxiv:comment>13 pages, 4 figures</arxiv:comment>\n <arxiv:primary_category term='math.NA'/>\n <author>\n <name>Andrew Qing He</name>\n </author>\n <author>\n <name>Wei Cai</name>\n </author>\n </entry>"
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