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

LEAPS-MPS：通过收敛分析、数据同化和深度学习探索各种亚网格尺度湍流模型

• 批准号: 2316894
• 负责人: Jing Tian
• 金额: $ 20.49万
• 依托单位: Towson University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316894&HistoricalAwards=false
• 关键词: LEAPS; MPS; Exploring; various; subgrid

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.

湍流是一组复杂的非线性现象，表现出流体的不稳定，不规则，似乎是随机和混乱的特征动作。它发生在各种系统中，涉及大气，海洋，空气动力学和技术。众所周知，湍流流动的研究非常重要和挑战。由于三维Navier-Stokes方程的某些存在和独特性的问题仍然尚未完全确立，但湍流建模目前提供了最佳的定性，在许多情况下，甚至在许多情况下甚至针对许多应用中的许多问题都提供了量化措施。最近，通过平均过程生成，开发了各种亚电网尺度湍流模型，并取得了良好的成功。这些模型不仅捕获了流动的大规模动力学，而且还提供了烟道物理学的“未解决”的小规模表示以及对平均方程的可靠闭合模型。此外，它们具有良好的分析，经验和计算特性，例如全球规则性，以及与从湍流通道和管道等示例中收集的经验数据良好的匹配。因此，就数学严谨性和现实世界应用而言，研究这些模型将是有益且有效的。该项目的目的是从基本数学研究和应用的角度研究这些亚网络湍流模型。该项目将通过参与无障碍研究项目，对本科生和研究生，尤其是来自代表性群体不足的群体的研究生产生重大影响。这还将通过与不同职业阶段的研究人员合作，建立数学系的研究能力和现代产品，以满足数据科学专业知识的区域需求并在当地社区提供教育经验，从而为PI建立强大的研究议程。在该项目中，将进行融合分析，数据同化算法和深度学习计算的综合工作，以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。这将有助于我们探索他们与全球规则性问题的有趣联系。接下来，PI将将数据同化算法应用于这些亚网格量表模型。 PI计划为这些模型构建一个数据同化系统，证明解决方案的存在，并显示了这些模型的数据同步解决方案与三维域上Navier-Stokes方程的弱解决方案的收敛性。作为目标，PI将通过确定图及其计算进行深度学习进行参数估计。 PI旨在通过确定地图和使用神经网络计算严格的框架。 PI和她的团队利用最近深度学习技术的计算优势，希望能够治疗传统数值方法面临障碍（例如维度和复杂几何形状的诅咒）的某些情况。该项目将有助于提供各种子网格规模的湍流模型与Navier-Stokes方程之间的启发性联系，并提高我们对湍流的理解。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子的优点和更广泛的影响来评估NSF的法定任务。