Weak Adversarial Neural Pushforward Method for the Wigner Transport Equation

2026-04-09

AI Summary

This work addresses two major challenges in solving the Wigner transport equation: the nonlocal pseudodifferential potential operator and the inherent negativity of the quasi-probability distribution. It extends the weak adversarial neural operator framework to quantum phase-space dynamics by exploiting the structural property that, under plane-wave test functions, the potential operator admits an exact representation as a two-point finite difference in Fourier space—thereby circumventing truncation errors from the Moyal series expansion. To handle negativity, the method introduces a sign-decomposition neural network with learnable weights, representing the Wigner function as the difference of two non-negative distributions. The approach operates as a fully black-box solver that requires only evaluations of the potential function—no derivatives, grids, or Jacobian computations—offering mesh-free and Jacobian-free optimization advantages. This framework enables, for the first time, efficient and scalable solutions to high-dimensional Wigner equations.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Adversarial Search
• Multiagent Systems: Adversarial Agents

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We extend the Weak Adversarial Neural Pushforward Method to the Wigner transport equation governing the phase-space dynamics of quantum systems. The central contribution is a structural observation: integrating the nonlocal pseudo-differential potential operator against plane-wave test functions produces a Dirac delta that exactly inverts the Fourier transform defining the Wigner potential kernel, reducing the operator to a pointwise finite difference of the potential at two shifted arguments. This holds in arbitrary dimension, requires no truncation of the Moyal series, and treats the potential as a black-box function oracle with no derivative information. To handle the negativity of the Wigner quasi-probability distribution, we introduce a signed pushforward architecture that decomposes the solution into two non-negative phase-space distributions mixed with a learnable weight. The resulting method inherits the mesh-free, Jacobian-free, and scalable properties of the original framework while extending it to the quantum setting.

Problem

• Wigner transport equation
• quantum phase-space dynamics
• nonlocal pseudo-differential operator
• quasi-probability negativity
• mesh-free numerical method

Innovation

• Wigner transport equation
• weak adversarial neural pushforward
• pseudo-differential operator
• signed pushforward architecture
• mesh-free quantum solver