Interplay Between Data and Partial Differential Equation Models Through the Lens of Kinetic Equations

由众多相互作用的粒子组成的系统普遍存在于各个领域。例如，我们呼吸的空气由多种分子组成，如相互作用的氮和氧，等离子体聚变能量依赖于大量相互作用的等离子体粒子，半导体涉及相互作用的离子和电子的流动。对这些相互作用的粒子系统的全面研究被称为动力学理论的普遍数学框架所包含。这一理论是理解和解决这些领域的工程挑战的基本基础。这个研究项目的主要焦点是通过一种结合了微分方程式分析和数据科学技术的综合方法来研究动力学理论。通过这样做，该项目的目的不仅是解开所涉及的方程的数学性质，而且还通过整合实验数据来准确地确定参数值。除了促进我们对纯数学的理解外，该项目还通过为在国家实验室和等离子体聚变能源行业进行的一系列特定实验提供严格的数学证明，从而具有显著的社会效益。作为这项努力的一部分，包括两名研究生和一名博士后在内的早期职业研究人员将接受培训，他们都属于STEM领域中代表性较低的群体，这将有助于促进多样化的劳动力。首先，将利用贝叶斯抽样和偏微分方程约束优化等数据科学工具来推断动力学方程中的未知参数。这自然属于逆问题的框架，其目的是通过以非侵入性的方式观察某些特征来确定系统可能的动态。其次，研究人员将探索应用动力学理论工具，特别是平均场理论和梯度流分析，来分析涉及多粒子系统模拟的机器学习算法。通过这些共同努力，该项目不仅旨在推动由智力好奇心驱动的数学研究，还寻求通过动力学理论的视角推动数学、计算机科学和工程学的边界，以实现综合科学进步。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。

Systems composed of numerous interacting particles are ubiquitous in various domains. For instance, the air we breathe consists of a multitude of molecules, such as interacting nitrogen and oxygen, plasma fusion energy relies on large quantities of interacting plasma particles, and semiconductors involve the flow of interacting ions and electrons. The comprehensive study of these interacting particle systems is encompassed by the universal mathematical framework known as kinetic theory. This theory serves as the fundamental basis for understanding and tackling engineering challenges in these fields. The primary focus of this research project is to investigate kinetic theory through an integrated approach that combines differential-equation analysis with data science techniques. By doing so, the project aims not only to unravel the mathematical properties of the equations involved but also to accurately determine parameter values by integrating experimental data. In addition to advancing our understanding of pure mathematics, this project holds significant societal benefits by providing rigorous mathematical justifications for a specific set of experiments conducted in national labs and the plasma fusion energy industry. As part of this endeavor, early career researchers, including two graduate students and one postdoc, will receive training, all of whom belong to underrepresented groups in STEM fields, which will help to promote the diverse workforce.The investigator will adopt two approaches. Firstly, data science tools, such as Bayesian sampling and PDE-constrained optimization, will be utilized to infer unknown parameters in kinetic equations. This falls naturally into the framework of inverse problems, where the aim is to determine the possible dynamics of a system by observing certain features in a non-intrusive manner. Secondly, the investigator will explore the application of kinetic theory tools, particularly mean-field theory and gradient flow analysis, to analyze machine learning algorithms that involve the simulation of many-particle systems. Through these combined efforts, the project not only aims to advance mathematical studies driven by intellectual curiosity but also seeks to push the boundaries of mathematics, computer science, and engineering for integrated scientific progress through the lens of kinetic theory.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.