Optimal Transport Applications to Probability, Machine Learning, and Kinetic Theory

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The project aims to transfer recently developed insights in mathematical analysis to other areas of interest including stochastic modelling, artificial intelligence, and kinetic theory. Stochastic models have recently become ubiquitous in physical science and social sciences to describe phenomena ranging from the evolution of political opinions to policy-driven segregation in urban environments. Artificial intelligence is a fast-growing field, relying on algorithms that have not yet been thoroughly studied mathematically. Kinetic theory has been studied in the context of many important applications such as space shuttle design, and it has become more relevant due to efforts to develop clean energy fusion reactors. The project will focus on developing mathematical frameworks and advancing the state of the art in those important fields. The project will also provide research training opportunities for graduate students. The project aims to develop mathematical frameworks for particle interactions, machine learning algorithms, and kinetic theory. The investigator will exploit theories developed in optimal mass transportation and gradient flows in metric spaces for complex dynamics by studying the associated free energy. For stochastic models, in particular weakly interacting diffusions, the project aims to develop a variational structure capturing the effect of phase transitions. For artificial intelligence, the project will focus on obtaining mean-field limits of parameter training and providing the functional structure of successful algorithms, such as Wasserstein generative adversarial network (GAN) and AlphaGo Zero. For kinetic theory, the project will exploit newly developed gradient flow formulations for the Landau and Boltzmann equations to obtain new insight into the behavior of these models.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.