Structure-preserving machine learning moment closures for kinetic equations

动力学理论从统计的角度描述动力系统的行为。它在许多领域有着广泛的应用，包括超声速流动、微机电系统、非常规气藏、空间飞行器再入问题和核聚变。由于这些模型的高维性，有效的模拟是一个长期的挑战，这限制了他们的应用到现实世界的问题。该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，预计将在机器学习（ML）的帮助下捕获物理学并作为良好的替代品。这将为物理和工程中的非平衡现象的建模和模拟提供一个强有力的工具。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部门提供课程开发。本研究的主要目标是开发具有可证明数学结构的鲁棒，准确和高效的ML矩模型。该项目的重点是如何保留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，对数值模拟的鲁棒性也至关重要。研究了以下思想和方法：（1）基于对称化器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，确保矩模型初始边值问题适定性的必要条件;（3）具有广义数据驱动矩的双曲性的ML模型。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

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.