Structure-preserving machine learning moment closures for kinetic equations

动力学理论从统计学的角度描述了动力系统的行为。它在超音速流动、微机电系统、非常规气藏、航天器再入问题、核聚变等领域有着广泛的应用。由于这类模型的高维性，有效的仿真是一个长期存在的挑战，这限制了它们在现实世界问题中的应用。这项研究项目将通过开发简化模型来近似动力学方程来解决这一挑战。这些模型被称为矩模型，有望在机器学习(ML)的帮助下捕捉物理学并作为很好的替代品。这将为物理和工程中的非平衡现象的建模和仿真提供有力的工具。该项目将为对计算数学感兴趣的研究生和本科生提供研究机会，并为PI系提供课程开发。本研究的主要目标是开发具有可证明的数学结构的稳健、准确和高效的最大似然矩模型。该项目的重点是如何保持ML矩模型的双曲性结构。双曲性与一阶偏微分方程组的适定性密切相关，对于稳健的数值模拟也是至关重要的。将研究以下思想和方法：(1)基于对称化的方法和基于特征值的方法，通过利用ML矩模型的代数结构在多维情况下保持模型的双曲性；(2)学习边界条件的ML方法，其确保矩模型的初始边值问题的适定性的必要条件；(3)由广义数据驱动矩强制的具有双曲性的ML模型。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Kinetic theory describes the behaviors of dynamic systems from a statistical point of view. It has wide applications in many fields, including supersonic flows, microelectromechanical systems, unconventional gas reservoirs, space vehicle re-entry problems, and nuclear fusion. Because of the high dimensionality of such models, efficient simulation is a long-standing challenge, which limits their applications to real-world problems. This research project will address this challenge by developing reduced models to approximate the kinetic equations. These models, called moment models, are expected to capture the physics and serve as good surrogates with the aid of machine learning (ML). This will provide a powerful tool in the modeling and simulation of non-equilibrium phenomena in physics and engineering. The project will provide research opportunities for graduate and undergraduate students who are interested in computational mathematics, and provide curriculum development in the PI's department.The primary objective of this research is to develop robust, accurate, and efficient ML moment models with some provable mathematical structures. The project focuses on how to preserve the hyperbolicity structure of the ML moment models. The hyperbolicity is closely related to the well-posedness of the first-order system of partial differential equations and is also vitally important for robust numerical simulations. The following ideas and methodologies will be investigated: (1) a symmetrizer-based approach and an eigenvalue-based approach that preserve the hyperbolicity of the model in multidimensional cases by exploiting the algebraic structure of the ML moment model; (2) a ML approach to learning boundary conditions that ensures necessary conditions for the well-posedness of the initial boundary value problem for the moment model; (3) a ML model with hyperbolicity enforced by generalized data-driven moments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.