Asymptotic-preserving neural networks for the semiconductor Boltzmann equation and its application on inverse problems

TL;DR

This paper introduces Asymptotic-Preserving Neural Networks (APNNs) for efficiently solving forward and inverse problems related to the semiconductor Boltzmann equation, especially in multiscale and data-sparse scenarios.

Contribution

The paper develops a novel APNN framework that incorporates asymptotic-preserving properties into the learning process for multiscale Boltzmann equations.

Findings

Effective in handling multiscale regimes

Accurate inverse problem solutions with sparse data

Validated through numerical experiments

Abstract

In this paper, we develop the Asymptotic-Preserving Neural Networks (APNNs) approach to study the forward and inverse problem for the semiconductor Boltzmann equation. The goal of the neural network is to resolve the computational challenges of conventional numerical methods and multiple scales of the model. To guarantee the network can operate uniformly in different regimes, it is desirable to carry the Asymptotic-Preservation (AP) property in the learning process. In a micro-macro decomposition framework, we design such an AP formulation of loss function. The convergence analysis of both the loss function and its neural network is shown, based on the Universal Approximation Theorem and hypocoercivity theory of the model equation. We show a series of numerical tests for forward and inverse problems of both the semiconductor Boltzmann and the Boltzmann-Poisson system to validate the effectiveness of our proposed method, which addresses the significance of the AP property when dealing with inverse problems of multiscale Boltzmann equations especially when only sparse or partially observed data are available.