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

该项目旨在将最近在数学分析方面发展起来的见解转移到其他感兴趣的领域，包括随机建模、人工智能和动力学理论。随机模型最近在自然科学和社会科学中变得无处不在，用于描述从政治观点的演变到城市环境中政策驱动的隔离等现象。人工智能是一个快速发展的领域，它所依赖的算法尚未得到彻底的数学研究。动力学理论已经在许多重要应用的背景下进行了研究，例如航天飞机的设计，并且由于开发清洁能源聚变反应堆的努力，它变得更加相关。该项目将侧重于发展数学框架和推进这些重要领域的最新技术。该项目还将为研究生提供研究培训机会。该项目旨在为粒子相互作用、机器学习算法和动力学理论开发数学框架。研究者将通过研究相关的自由能，利用在复杂动力学的度量空间中最优质量运输和梯度流动的理论。对于随机模型，特别是弱相互作用扩散，该项目旨在开发一种捕捉相变效应的变分结构。对于人工智能，该项目将专注于获得参数训练的平均场极限，并提供成功算法的功能结构，如Wasserstein生成对抗网络（GAN）和AlphaGo Zero。对于动力学理论，该项目将利用新开发的朗道和玻尔兹曼方程的梯度流动公式，以获得对这些模型行为的新见解。该奖项反映了美国国家科学基金会的法定使命，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，被认为值得支持。

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.