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

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

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.