LEAPS-MPS: Exploring various subgrid scale turbulence models via convergence analysis, data assimilation and deep learning

湍流是一组具有非定常、不规则、看似随机和混沌特征的流体运动的复杂非线性现象。它发生在各种系统中，包括大气、海洋、空气动力学和技术。众所周知，湍流流体流动的研究是非常重要和具有挑战性的。由于经过一个世纪的努力，三维Navier-Stokes方程解的存在性和唯一性问题仍未完全建立，湍流建模目前为应用中的许多问题提供了最好的定性，在许多情况下甚至是定量的测量方法。近年来，通过平均过程生成各种亚网格尺度湍流模型，并取得了较好的效果。这些模型不仅捕获了流动的大尺度动力学，而且还提供了流体物理的“未解决”的小尺度表示以及平均方程的可靠闭合模型。此外，它们具有良好的分析性、经验性和计算性，如全局规律性和与湍流通道和管道等实例收集的经验数据的良好匹配。因此，就数学严谨性和实际应用而言，研究这些模型将是有益和有效的。本项目旨在从基础数学研究和应用的角度对这些亚网格湍流模型进行研究。该项目将对本科生和研究生产生重大影响，特别是那些来自代表性不足群体的学生，通过他们参与无障碍研究项目。这也将通过与不同职业阶段的研究人员合作，为PI建立一个强大的研究议程，建立数学系的研究能力和课程，以满足区域对数据科学专业知识的需求，并在当地社区提供教育经验。本项目将综合收敛分析、数据同化算法和深度学习计算，研究各种亚网格尺度湍流模型。PI将首先探索与Navier-Stokes方程相关的这些模型之间的关系和出现。这将有助于我们探索它们与全球规律性问题之间的有趣联系。接下来，PI将对这些子网格尺度模型应用数据同化算法。PI计划为这些模型建立一个数据同化系统，证明解的存在性，并证明这些模型的数据同化解在三维域上收敛于Navier-Stokes方程的弱解。作为目标，PI将通过确定映射及其使用深度学习的计算进行参数估计。PI的目标是通过确定地图和使用神经网络计算来开发一个严格的框架。利用最近深度学习技术的计算优势，PI和她的团队希望能够处理传统数值方法面临障碍的一些情况，例如维度诅咒和复杂几何形状。这个项目将有助于在各种亚网格尺度湍流模型和Navier-Stokes方程之间提供一个启发性的联系，并提高我们对湍流的理解。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。

Turbulence is a set of complex nonlinear phenomena showing unsteady, irregular, seemingly random and chaotic characteristic motions of fluids. It happens in various systems involving the atmosphere, ocean, aerodynamics, and technology. The study of turbulent fluid flow is well known to be highly important and challenging. Since some issues of existence and uniqueness of solutions of the three-dimensional Navier-Stokes equations are still not yet fully established despite century-long efforts, turbulence modeling currently provides the best qualitative and, in many cases, even quantitative measures for many problems in applications. Recently, generating through an averaging process, various subgrid scale turbulence models were developed with good success. These models not only capture the large-scale dynamics of the flow, but also provide “unresolved” small-scale representations of the physics of fluids as well as reliable closure models to the averaged equations. Moreover, they have nice analytical, empirical and computational properties, such as global regularity and good matching with empirical data collected from examples such as turbulent channels and pipes. Therefore, studying these models will be beneficial and effective as far as both mathematical rigors and real-world applications are concerned. This project aims to study these subgrid turbulence models from the point of view of both basic mathematical research and applications. The project will have significant impacts on both undergraduate and graduate students, particularly those from underrepresented groups, through their participation in accessible research projects. This will also establish a strong research agenda for the PI via working with different career-stage researchers, building the research capability and curricular offerings of the Department of Mathematics to fulfill regional needs for data science expertise and offering educational experiences in the local community. In this project, a synthesizing effort of convergence analysis, data assimilation algorithm and deep learning computation will be made to study various subgrid scale turbulence models. The PI will first explore the relationship between, and the emergence of, these models in association with the Navier-Stokes equations. This will help us explore their intriguing connections to the global regularity problem. Next, the PI will apply the data assimilation algorithm to these subgrid scale models. The PI plans to build a data assimilation system for these models, prove the existence of solutions, and show convergence of the data-assimilated solutions of these models to the weak solutions of the Navier-Stokes equations on a three-dimensional domain. As a target, the PI will conduct parameter estimation via the determining map and its computation using deep learning. The PI aims to develop a rigorous framework via the determining map and computing it using neural networks. Exploiting the computational advantages of recent deep learning techniques, the PI and her team hope to be able to treat some cases where the traditional numerical methods face hurdles such as the curse of dimensionality and complex geometries. This project will help provide an illuminating link between various subgrid scale turbulence models and the Navier-Stokes equations and improve our understanding of turbulence.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.