标题： 基于机器学习的线性玻尔兹曼方程不确定性矩闭合模型 显示英文标题

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

摘要： 玻尔兹曼方程是动力学理论中的基本方程，它在微观粒子动力学和宏观连续介质力学之间架起了一座桥梁。然而，由于矩层次结构的固有非闭合性，从玻尔兹曼方程导出封闭的宏观矩系统仍然是一个长期存在的挑战。在本文中，我们提出了一种基于机器学习（ML）的线性玻尔兹曼方程的矩闭合模型，解决了确定性和随机设置下的问题。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，为了确保全局双曲性和稳定性，我们推导并应用了强制系统可对称双曲性的约束。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使用确定性情况下方法的系统。通过几个数值实验展示了我们的基于机器学习的矩闭合模型在具有或不具有不确定性的情况下对线性玻尔兹曼方程的有效性和准确性。 显示英文摘要

摘要： The Boltzmann equation, a fundamental equation in kinetic theory, serves as a bridge between microscopic particle dynamics and macroscopic continuum mechanics. However, deriving closed macroscopic moment systems from the Boltzmann equation remains a long-standing challenge due to the intrinsic non-closure of the moment hierarchy. In this paper, we propose a machine learning (ML)-based moment closure model for the linear Boltzmann equation, 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 are shown to demonstrate the effectiveness and accuracy of our ML-based moment closure model for the linear Boltzmann equation with or without uncertainties.