A Bi-fidelity based asymptotic-preserving neural network for the semiconductor Boltzmann equation and its inverse problem

Abstract: This paper introduces a Bi-fidelity Asymptotic-Preserving Neural Network (BI-APNNs) framework, designed to efficiently solve forward and inverse problems for the semiconductor Boltzmann equation. Our approach builds upon the Asymptotic-Preserving Neural Network (APNNs) methodology \cite{APNN-transport}, which employs a micro-macro decomposition to handle the model's multiscale nature. We specifically address a key bottleneck in the original APNNs: the slow convergence of the macroscopic density $ρ$ in the near fluid-dynamic regime, i.e., for small Knudsen numbers $\varepsilon$. The core innovation of BI-APNNs is a novel bi-fidelity decomposition of the macroscopic quantity $ρ$, which accurately approximates the true density at small $\varepsilon$, and can be efficiently pre-trained. A separate and more compact neural network is then tasked with learning only the minor correction term, $ρ_{\text{corr}}$. This strategy not only significantly {\it accelerates} the training convergence but also improves the accuracy of the forward problem solution, particularly in the challenging fluid-dynamic limit. Meanwhile, we demonstrate through extensive numerical experiments that our new BI-APNNs yields substantially more accurate and robust results for inverse problems compared to the standard APNNs. Validated on both the semiconductor Boltzmann and the Boltzmann-Poisson systems, our work shows that the bi-fidelity formulation is a powerful enhancement for tackling multiscale kinetic equations, especially when dealing with inverse problems constrained by partial observation data.