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.

科学和数据分析中动态行为的众多重要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。物理和信息科学领域的许多系统倾向于最大化熵。深度学习算法是通过优化聚类和分类大数据集的参数来训练的。该项目将在许多学科的研究人员中，在几种当前兴趣的几种模型中，我们对最佳原理和动态的数学理解。这些范围从流体动力学和网络路由到统计抽样和数据科学再到气溶胶物理和动物生态学。最佳运输理论将以新颖的方式使用，以模拟流体混合物动力学并了解流体表面奇异性如何形成。将研究梯度下降技术，以分析和改善高维统计抽样和波形计算的收敛性。合并聚类模型的新型动力学现象将在与气雾剂颗粒增长和金融市场中信息共享有关的模型中寻求。这些调查将刺激年轻的研究人员和学生参与，并将导致在会议，研究机构，研究机构，研讨会和讲座系列中进行分解。尤其是，该项目的研究将着重于将各种分析的想法从各种兴趣中置于当前兴趣的几个特定主题上：（1）（1）通过构建不可压缩的多种形式的多种型号的成型，该组合是如何进行多个型成型的成型，该组合的成型是多种型号的成型。可行的计算，并可以对最佳动态途径进行精确表征；（2）在涉及从角落扩张的情况下，通过对简单的大地测量流的新型扰动分析，证明了不可压缩流体表面上的奇异性的形成；（3）建立梯度样流量的收敛来解释相干状态的形成并改善统计抽样，通过开发在无限二维非局部模型中使用lojasiewicz估计值；（4）在聚集和分手的动力学模型中识别可稳定状态和非平凡的时间动力学，这些模型缺乏详细的体重结构，该结构将推动Syst EM达到均衡。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子的智力和更广泛的影响来通过评估来进行评估的支持。