Weak Adversarial Neural Pushforward Method for the Wigner Transport Equation

TL;DR

This paper extends a neural pushforward method to solve the Wigner transport equation, enabling scalable, mesh-free quantum phase-space simulations without truncating series or requiring derivatives of the potential.

Contribution

It introduces a novel structural insight that simplifies the operator, handles negativity via a signed architecture, and extends the framework to quantum systems without truncation or derivative info.

Findings

Exact inversion of the Wigner potential kernel in arbitrary dimensions.

Decomposition of the solution into non-negative distributions to handle negativity.

Scalable, mesh-free, Jacobian-free quantum phase-space simulation method.

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.