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

在科学和数据分析中，许多重要的动态行为模型的数学结构从根本上与最优性有关。随着时间的推移，流体运动优化了动能。物理和信息科学中的许多系统倾向于使熵最大化。深度学习算法通过优化大数据集聚类和分类的参数来训练。该项目将提高我们对当前许多学科研究人员感兴趣的几个模型中最优性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抽样和数据科学到气溶胶物理学和动物生态学。最优输运理论将以一种新颖的方式用于模拟流体混合动力学和理解流体表面奇点是如何形成的。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。在与大气动力学中的气溶胶粒子增长和金融市场信息共享相关的模型中，将寻求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引入当前感兴趣的几个具体主题：(1)模拟不可压缩流体如何通过熵正则化的多边际最优输运公式进行最佳混合，这应该使数值计算可行，并可能使最优动态路径的精确表征；(2)通过对简单测地线流动的新颖摄动分析，证明了不可压缩流体在从一个角落膨胀的情况下表面奇点的形成；(3)通过在无限维非局部模型中使用Lojasiewicz估计，建立类梯度流的收敛性，以解释相干态的形成并改进统计抽样；(4)识别聚集和破裂动力学模型中的亚稳态和非平凡时间动力学，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 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.