AP-MIONet: Asymptotic-preserving multiple-input neural operators for capturing the high-field limits of collisional kinetic equations

TL;DR

This paper introduces AP-MIONet, a novel neural operator approach that effectively solves high-field kinetic equations like VPFP and Boltzmann, handling variable parameters without explicit local equilibrium assumptions.

Contribution

The paper presents a new asymptotic-preserving multiple-input DeepONet architecture tailored for high-field kinetic equations, overcoming challenges related to additional parameters and non-equilibrium states.

Findings

Accurately captures high-field regimes in kinetic equations.

Handles variable electric field parameters effectively.

Preserves mass conservation without explicit local equilibrium.

Abstract

In kinetic equations, external fields play a significant role, particularly when their strength is sufficient to balance collision effects, leading to the so-called high-field regime. Two typical examples are the Vlasov-Poisson-Fokker-Planck (VPFP) system in plasma physics and the Boltzmann equation in semiconductor physics. In this paper, we propose a generic asymptotic-preserving multiple-input DeepONet (AP-MIONet) method for solving these two kinetic equations with variable parameters in the high-field regime. Our method aims to tackle two major challenges in this regime: the additional variable parameters introduced by electric fields, and the absence of an explicit local equilibrium, which is a key component of asymptotic-preserving (AP) schemes. We leverage the multiple-input DeepONet (MIONet) architecture to accommodate additional parameters, and formulate the AP loss function by incorporating both the mass conservation law and the original kinetic system. This strategy can avoid reliance on the explicit local equilibrium, preserve the mass and adapt to non-equilibrium states. We demonstrate the effectiveness and efficiency of the proposed method through extensive numerical examples.