Weak Adversarial Neural Pushforward Method for the Wigner Transport Equation
2026-04-09 • Machine Learning
Machine Learning
AI summaryⓘ
The authors extend a neural network method to solve the Wigner transport equation, which describes quantum systems in phase space. They found a new way to simplify a complex operator into a finite difference that doesn't need approximations or derivatives of the potential. To handle the fact that the Wigner distribution can have negative values, they designed a network that splits the solution into two positive parts combined by a learnable weight. This makes their method flexible, efficient, and applicable directly to quantum problems.
Wigner transport equationphase-space dynamicspseudo-differential operatorFourier transformWigner potential kernelMoyal seriesquasi-probability distributionpushforward methodneural networksquantum systems
Authors
Andrew Qing He, Wei Cai, Sihong Shao
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.