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化有关。随着时间的推移，流体运动会优化动能。物理和信息科学中的许多系统都倾向于使熵最大化。通过优化大数据集的聚类和分类参数来训练深度学习算法。这个项目将提高我们对最优性原理和动力学在几个模型中的数学理解，这些模型目前对许多学科的研究人员很感兴趣。这些学科的范围从流体动力学和网络布线，到统计采样和数据科学，再到气溶胶物理和动物生态学。最优输运理论将以一种新的方式用于模拟流体混合物的动力学，并了解流体表面奇点是如何形成的。将研究梯度下降技术，以分析和改进高维统计抽样和波形计算的收敛。将在与大气动力学中的气溶胶颗粒增长和金融市场信息共享有关的模型中寻找集群合并-分裂模型中的新的动力学现象。这些研究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与当前感兴趣的几个具体主题联系起来：(1)通过熵正则多边际最优传输公式模拟不可压缩流体如何最佳混合，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成为可能；(2)通过对简单测地线流的一种新的微扰分析，演示了在从拐角膨胀的情况下不可压缩流体表面奇点的形成：(3)通过发展无限维非局域模型中的Lojasiewicz估计，建立了梯度流的收敛，以解释相干态的形成和改进统计抽样；(4)在聚集和分解的动力学模型中识别亚稳态和非平凡的时间动力学，这些亚稳态和非平凡的时间动力学缺乏将系统推向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 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.