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

通过动力学方程的视角观察数据和偏微分方程模型之间的相互作用

• 批准号: 2308440
• 负责人: Qin Li
• 金额: $ 28.83万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2308440&HistoricalAwards=false
• 关键词: Interplay; Between; Data; Partial; Differential

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.

由许多相互作用粒子组成的系统在各个域中无处不在。例如，我们呼吸的空气由多种分子组成，例如相互作用的氮和氧，等离子体融合能依赖大量相互作用的血浆颗粒，半导体涉及相互作用的离子和电子的流动。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通用数学框架所包含。该理论是理解和应对这些领域的工程挑战的基本基础。该研究项目的主要重点是通过将差分方程分析与数据科学技术结合在一起的综合方法来研究动力学理论。通过这样做，该项目不仅旨在揭示所涉及的方程的数学特性，还旨在通过集成实验数据来准确确定参数值。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血浆融合能源行业进行的一组特定的实验提供严格的数学合理性，从而具有重大的社会利益。作为这项努力的一部分，包括两名研究生和一名博士后的早期职业研究人员将接受培训，所有这些研究人员均属于STEM领域中代表性不足的群体，这将有助于促进多样化的劳动力。研究人员将采用两种方法。首先，将利用数据科学工具（例如贝叶斯采样和PDE受限优化）来推断动力学方程中未知参数。这自然属于反问题的框架，在这种框架中，目的是通过以非侵入性方式观察某些特征来确定系统的可能动态。其次，研究者将探讨动力学理论工具的应用，尤其是均值场理论和梯度流分析，以分析涉及许多颗粒系统模拟的机器学习算法。通过这些结合的努力，该项目不仅旨在推进智力好奇心驱动的数学研究，而且还试图通过动力学理论的镜头来推动数学，计算机科学和工程学的界限，以综合科学进步。该奖项反映了NSF的法定任务，并通过使用基金会的Merit和Broadial and Imparia和广泛的评估来反映出值得评估的支持，并具有值得的支持。