Weak Adversarial Neural Pushforward Method for Boltzmann Equation

2026-08-07

AI Summary

This work addresses the challenge of efficiently solving the high-dimensional collision operator in the time-dependent Boltzmann equation by proposing a weak adversarial neural pushforward method. The approach integrates invertible neural networks with a weak-form variational principle, leveraging adversarial training to construct a pushforward mapping that inherently satisfies the constraints imposed by the Boltzmann equation. This enables direct generation of samples consistent with the solution distribution, thereby circumventing explicit evaluation of the computationally expensive high-dimensional collision integral. To the best of our knowledge, this is the first study to combine weak-form formulations with adversarial neural pushforward mechanisms for Boltzmann equation solvers. Numerical experiments demonstrate that the method achieves significant computational speedup while maintaining accuracy, highlighting its feasibility and scalability for high-dimensional kinetic simulations.

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.

Problem

• Boltzmann equation
• collision operator
• time-dependent
• weak formulation
• neural pushforward

Innovation

• weak adversarial learning
• neural pushforward
• Boltzmann equation
• invertible neural networks
• weak formulation