Weak Adversarial Neural Pushforward Method for Boltzmann Equation

Jenia Fardousi Koly; Andrew Qing He; Wei Cai

math.NAAug 7, 2026

Weak Adversarial Neural Pushforward Method for Boltzmann Equation

Authors: Jenia Fardousi Koly, Andrew Qing He, Wei Cai

Organizations: Department of Mathematics, Southern Methodist University, Dallas, Texas, USA

Abstract

In this paper, we extend a weak adversary neural network pushforward method for solving time dependent Boltzmann equation and a weak formulation of the collision operator is proposed where an invertible neural pushforward mapping is used to generating samples given by the distribution governed by the Boltzmann equation. The training of the pushforward mapping is learnt by enforcing the weak form of the Boltzmann equation. Numerical results have demonstrated the effectiveness of the proposed method.

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Apr 9, 2026 · quant-phJ/K move · Enter open · S save

Weak Adversarial Neural Pushforward Method for the Wigner Transport Equation

Andrew Qing He, Wei Cai, Sihong Shao

*Department of Mathematics, Southern Methodist University, Dallas, TX, USA. · †School of Mathematical Sciences, Peking University, Beijing, China.

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.

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Abstract

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