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

Liu Liu, Yating Wang, Xueyu Zhu and Zhenyi Zhu
arXiv.org
Cornell University
07/23/2024
DOI: 10.48550/arxiv.2407.16169
url
https://doi.org/10.48550/arxiv.2407.16169View
Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

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.
 Mathematics - Mathematical Physics Physics - Mathematical Physics

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