In this paper, we propose a machine learning (ML)-based moment closure model for the linearized Boltzmann equation of semiconductor devices, addressing both the deterministic and stochastic settings. Our approach leverages neural networks to learn the spatial gradient of the unclosed highest-order moment, enabling effective training through natural output normalization. For the deterministic problem, to ensure global hyperbolicity and stability, we derive and apply the constraints that enforce symmetrizable hyperbolicity of the system. For the stochastic problem, we adopt the generalized polynomial chaos (gPC)-based stochastic Galerkin method to discretize the random variables, resulting in a system for which the approach in the deterministic case can be used similarly. Several numerical experiments will be shown to demonstrate the effectiveness and accuracy of our ML-based moment closure model for the linear semiconductor Boltzmann equation with (or without) uncertainties.

翻译：本文针对半导体器件的线性化玻尔兹曼方程，提出了一种基于机器学习（ML）的矩封闭模型，该模型同时适用于确定性及随机性场景。我们的方法利用神经网络学习未封闭最高阶矩的空间梯度，通过自然的输出归一化实现有效训练。对于确定性问题，为确保系统的全局双曲性与稳定性，我们推导并应用了强制系统可对称化双曲性的约束条件。对于随机性问题，我们采用基于广义多项式混沌（gPC）的随机伽辽金方法对随机变量进行离散化，由此得到的系统可类似地应用确定性情形中的方法。我们将展示若干数值实验，以验证所提出的基于机器学习的矩封闭模型对于含（或不含）不确定性的线性半导体玻尔兹曼方程的有效性与准确性。