Weak Adversarial Neural Pushforward Method for the Wigner Transport Equation
This provides a novel computational tool for simulating quantum transport in arbitrary dimensions without derivative information, benefiting researchers in quantum mechanics and computational physics.
The paper extends the Weak Adversarial Neural Pushforward Method to solve the Wigner transport equation for quantum phase-space dynamics, achieving a mesh-free, Jacobian-free, and scalable approach by reducing the nonlocal potential operator to a pointwise finite difference. The method handles negative Wigner distributions via a signed pushforward architecture.
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.