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

TL;DR

The paper tackles high-field multiscale kinetic equations, focusing on the Vlasov–Poisson–Fokker–Planck system and the semiconductor Boltzmann equation. It introduces AP-MIONet, a multi-input neural operator framework that integrates a mass-conservation–driven AP loss to capture correct high-field limits without requiring an explicit local equilibrium. By learning three operators for the distribution, density, and potential via dedicated MIONets and enforcing macroscopic consistency, the method achieves accurate results across kinetic and high-field regimes, including non-degenerate isotropic/anisotropic and degenerate cases, with substantial inference speedups. The approach demonstrates formal AP properties and robust performance in diverse test cases, offering a scalable, mesh-free tool for multiscale plasma and semiconductor simulations.

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.

Figures (10)

• Figure 1: The architecture of (a) MIONet and (b) physics-informed MIONet (PI-MIONet). In subfigure (a), assume the operator $\mathcal{G}$ accepts $n$ input functions $\boldsymbol{u}_1, \cdots \boldsymbol{u}_n$ and is evaluated at coordinate $\boldsymbol{y}$. The MIONet architecture corresponding to $\mathcal{G}$ comprises $n+1$ sub-networks: the $n$ branch nets, each for extracting latent representations from their respective input function $\{\boldsymbol{u}_i\}_{i=1}^n$; and a single trunk net for extracting latent representations from the input coordinate $\boldsymbol{y}$ at which the output function is evaluated. A continuous and differentiable representation of the output function $\mathcal{G}(\boldsymbol{u}_1,\cdots, \boldsymbol{u}_n)(\boldsymbol{y})$ is obtained by integrating the latent representations extracted by all sub-networks through a dot product. In subfigure (b), to make the MIONet physics-informed, additional regularization mechanisms are incorporated via automatic differentiation for biasing the MIONet output to satisfy the governing PDE system, boundary condition (BC) and initial condition (IC).
• Figure 2: Schematic illustrations of the physics-informed MIONet (PI-MIONet) architecture generic to both the VPFP system and the semiconductor Boltzmann equation in the high-field regime. (a) The architecture of the MIONet with modified MLPs for the parameterized operator $\mathcal{F}_\theta$ within the PI-MIONet framework. (b) The overall architecture of the PI-MIONet.
• Figure 3: The architecture of the Asymptotic-Preserving Multiple-Input DeepONet (AP-MIONet) generic to both the VPFP system and the semiconductor Boltzmann equation in the high-field regime.
• Figure 4: Landau damping for the VPFP system solved by the AP-MIONet method in the kinetic and high-field regimes. Density $\rho$ and electric field $E$ plotted in $(t,x)\in[0,5]\times[0, 2\pi/k]$, and electric energy plotted in $t\in[0,5]$ for a representative input function. Penalty $\lambda_1=100$ and other penalties are set to $1$ for kinetic regime; Penalty $\lambda_1=500$ and other penalties are set to $1$ for the high-field regime.
• Figure 5: Double peak instability for the non-degenerate isotropic semiconductor Boltzmann equation solved by the AP-MIONet method in the kinetic and high-field regimes. Density $\rho$ and electric field $E$ plotted in $(t,x)\in[0,1]\times[0, 2\pi/k]$ for a representative input function. Penalty $\lambda_1=20$ and other penalties are set to $1$ for the kinetic regime; Penalty $\lambda_1=100$ and other penalties are set to $1$ for the high-field regime.

Theorems & Definitions (3)

• Remark 1
• Definition 1
• Remark 2