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.

翻译：在动力学方程中，外部场起着重要作用，尤其当其强度足以平衡碰撞效应时，会引发所谓的高场区域。两个典型例子是等离子体物理中的Vlasov-Poisson-Fokker-Planck（VPFP）系统与半导体物理中的Boltzmann方程。本文提出了一种通用的保渐近多输入DeepONet（AP-MIONet）方法，用于求解高场区域中具有可变参数的这两类动力学方程。我们的方法旨在应对该区域中的两个主要挑战：由电场引入的额外可变参数，以及作为保渐近（AP）格式关键组件的显式局部平衡态的缺失。我们利用多输入DeepONet（MIONet）架构来容纳额外参数，并通过结合质量守恒定律与原动力学系统构建AP损失函数。该策略可避免对显式局部平衡态的依赖，保持质量守恒并适应非平衡态。我们通过大量数值算例验证了所提方法的有效性与高效性。