Collaborative Research: Dynamics, singularities, and variational structure in models of fluids and clustering

合作研究：流体和聚类模型中的动力学、奇点和变分结构

基本信息

批准号：
2106988
负责人：
Jian-Guo Liu
金额：
$ 23万
依托单位：
Duke University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2106988&HistoricalAwards=false
关键词：
Collaborative Research Dynamics singularities variational

项目摘要

The mathematical structure of numerous important models of dynamic behavior in science and data analysis is fundamentally related with optimality. Fluid motions optimize kinetic energy over time. Many systems in physical and information sciences tend to maximize entropy. Deep learning algorithms are trained by optimizing parameters for clustering and classifying big data sets. This project will improve our mathematical understanding of optimality principles and dynamics in several models of substantial current interest to researchers in a number of disciplines. These range from fluid dynamics and network routing to statistical sampling and data science to aerosol physics and animal ecology. Optimal transport theory will be used in a novel way to model fluid mixture dynamics and understand how fluid surface singularities can form. Gradient descent techniques will be investigated to analyze and improve the convergence of high-dimensional statistical sampling and wave-shape computations. Novel dynamical phenomena in merging-splitting models of clustering will be sought in models relevant to aerosol particle growth in atmospheric dynamics and the sharing of information in financial markets. These investigations will stimulate young researchers and students to participate, and will lead to results to be disseminated at conferences, research institutes, seminars, and lecture series.In particular, this project's research will focus on bringing ideas from variational analysis to bear upon several specific topics of current interest: (1) modeling how incompressible fluids may optimally mix through an entropy-regularized multi-marginal optimal transport formulation, which ought to make numerical computations feasible and may enable a precise characterization of optimal dynamic pathways; (2) demonstrating the formation of singularities on the surface of incompressible fluids in a scenario involving expansion from a corner, through a novel perturbation analysis of a simple geodesic flow; (3) establishing convergence of gradient-like flows to explain coherent-state formation and improve statistical sampling, by developing the use of Lojasiewicz estimates in infinite-dimensional nonlocal models; (4) identifying metastable states and nontrivial temporal dynamics in kinetic models of aggregation and breakup that lack a detailed-balance structure that would drive the syst em to equilibrium.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.

科学和数据分析中许多重要的动态行为模型的数学结构从根本上与最优性相关。流体运动随着时间的推移优化动能。物理和信息科学中的许多系统都倾向于最大化熵。深度学习算法是通过优化大数据集聚类和分类参数来训练的。该项目将提高我们对多个学科研究人员当前感兴趣的几种模型中的最优原理和动力学的数学理解。这些范围从流体动力学和网络路由到统计采样和数据科学，再到气溶胶物理学和动物生态学。最佳传输理论将以一种新颖的方式用于模拟流体混合物动力学并了解流体表面奇点如何形成。将研究梯度下降技术来分析和改进高维统计采样和波形计算的收敛性。将在与大气动力学中气溶胶颗粒生长和金融市场信息共享相关的模型中寻找聚类合并-分裂模型中的新动力学现象。这些调查将激励年轻的研究人员和学生参与，并将导致结果在会议、研究机构、研讨会和系列讲座中传播。特别是，该项目的研究将侧重于将变分分析的思想应用于几个具体的问题当前感兴趣的主题：（1）对不可压缩流体如何通过熵正则化多边际最佳输运公式进行最佳混合进行建模，这应该使数值计算变得可行，并可以实现最佳动态的精确表征途径； (2) 通过对简单测地线流的新颖扰动分析，展示在涉及从角部膨胀的情况下不可压缩流体表面奇点的形成； (3) 通过在无限维非局部模型中开发 Lojasiewicz 估计的使用，建立类梯度流的收敛性，以解释相干态的形成并改进统计采样； (4) 识别聚合和分解动力学模型中的亚稳态和非平凡的时间动力学，这些模型缺乏驱动系统达到平衡的详细平衡结构。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准。