标题： 基于机器学习的考虑不确定性的半导体玻尔兹曼方程的矩闭合模型 显示英文标题

标题： Machine learning-based moment closure model for the semiconductor Boltzmann equation with uncertainties

摘要： 在本文中，我们提出了一种基于机器学习（ML）的矩闭合模型，用于半导体器件的线性化玻尔兹曼方程，在确定性和随机设置中均进行了研究。 我们的方法利用神经网络来学习未闭合的最高阶矩的空间梯度，通过自然输出归一化实现有效的训练。 对于确定性问题，为确保全局双曲性和稳定性，我们推导并应用了使系统满足可对称双曲性的约束条件。 对于随机问题，我们采用了基于广义多项式混沌（gPC）的随机伽辽金方法来离散随机变量，从而得到一个可以类似地使用确定性情况方法的系统。 将展示几个数值实验，以证明我们的基于ML的矩闭合模型在具有（或不具有）不确定性的线性半导体玻尔兹曼方程中的有效性和准确性。 显示英文摘要

摘要： 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.