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

该项目旨在将数学分析方面最新发展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，用来描述从政治观点的演变到城市环境中政策驱动的隔离等一系列现象。人工智能是一个快速增长的领域，依赖于尚未在数学上进行彻底研究的算法。动力学理论在航天飞机设计等许多重要应用中得到了研究，由于开发清洁能源聚变反应堆的努力，它变得更加相关。该项目将侧重于开发数学框架，并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。研究人员将通过研究相关的自由能，在复杂动力学的度规空间中利用最优质量传输和梯度流中的理论。对于随机模型，特别是弱相互作用的扩散，该项目的目标是开发一种捕捉相变影响的变分结构。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生成性对手网络(GAN)和AlphaGo Zero。对于动力学理论，该项目将利用最新开发的Landau和Boltzmann方程的梯度流公式，以获得对这些模型行为的新见解。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。

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.