# Weak Adversarial Neural Pushforward Method for the Wigner Transport Equation

arXiv: https://arxiv.org/abs/2604.08763 · HTML: https://mlsanity.com/paper/2604.08763/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 13 references that are in the map.
- First posted on arXiv 9 Apr 2026.

## Builds on (most cited first, 1 of 1)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)

## Abstract (arXiv)

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.