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Controlling Geometry: Applications in Physics, Biology, and Manifold Learning

控制几何：在物理、生物学和流形学习中的应用

基本信息

批准号：
2111474
负责人：
Shawn Walker
金额：
$ 33万
依托单位：
Louisiana State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111474&HistoricalAwards=false
关键词：
Controlling Geometry Applications Physics Biology

项目摘要

New materials can be created by directing the assembly of fine-scale structures into optimal, desired patterns. This project is about controlling the shapes of things. Controlling the shapes of droplets can yield new micro-fluidic devices for bio-technology. Understanding the shape and curvature of bio-membranes (e.g. cell membranes) can give new insight into how cells move and function. And human understanding and meaning can be extracted from high dimensional data if it is properly "unfolded." The goal of this research project is to create new mathematical methods/algorithms to guide self-organization, material design, and learn from high dimensional data. This research will lay the groundwork for the optimal control of moving shapes and geometries. Part of this project involves interacting with elementary and middle school students to highlight the importance of geometry in applications through the PI's "sit-with-a-scientist" program. The program provides an informal atmosphere, with hands-on activities, to motivate students, especially minorities and under-represented groups, to pursue STEM.The research objective is to create new mathematical techniques and numerical methods for self-organization. Some examples are self-assembly, controlling droplet shape (micro-fluidics), folding biomembranes, and data analysis/visualization through non-linear manifold reduction. These new methods will open new frontiers of material design, enable unprecedented control of physical phenomena, yield new understanding in micro-biology, and reign in "big data" so it can be directly visualized. The research will create the first numerical scheme for the singular Maier-Saupe potential for the Q-tensor-valued solution of the Landau-de Gennes (LdG) model. We also develop and analyze, unfitted finite element methods (FEMs) for LdG, on variable domains, that connect with the following items. We will create methods for controlling the time-dependent evolution of geometric structures in physics and engineering problems, e.g. in liquid crystals and liquid droplet shape. Design new methods for modeling and simulating bio-membranes that well approximate full curvature information (i.e. the full shape operator) using a surface finite element method. Moreover, we will extend our bio-membrane surface FEM techniques to do non-linear dimension reduction of high dimensional data to a low-dimensional space (for data analytics and visualization). Other aspects of the research will create open source software for the methods developed here, using both the PI's own packages, FELICITY and AHF, and other open-source options (e.g. Firedrake). In addition, the PI will educate elementary and middle school students using his "sit-with-a-scientist" program (mentioned above).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.

通过将精细结构组装成最佳的、所需的图案，可以创造出新材料。这个项目是关于控制事物的形状。控制液滴的形状可以产生用于生物技术的新的微流体装置。了解生物膜（例如细胞膜）的形状和曲率可以为细胞如何移动和发挥功能提供新的见解。如果正确地“展开”，人类的理解和意义就可以从高维数据中提取出来。该研究项目的目标是创建新的数学方法/算法来指导自组织、材料设计并从高维数据中学习。这项研究将为运动形状和几何形状的优化控制奠定基础。该项目的一部分涉及与中小学生互动，通过 PI 的“与科学家坐在一起”项目来强调几何在应用中的重要性。该项目提供了一个非正式的氛围和实践活动，以激励学生，特别是少数族裔和代表性不足的群体，追求 STEM。研究目标是为自组织创造新的数学技术和数值方法。一些例子包括自组装、控制液滴形状（微流体）、折叠生物膜以及通过非线性流形简化进行数据分析/可视化。这些新方法将开辟材料设计的新领域，实现对物理现象的前所未有的控制，产生对微生物学的新理解，并统治“大数据”，使其可以直接可视化。该研究将为 Landau-de Gennes (LdG) 模型的 Q 张量值解的奇异 Maier-Saupe 势创建第一个数值方案。我们还在变量域上开发和分析 LdG 的未拟合有限元方法 (FEM)，这些方法与以下项目相关。我们将创建控制物理和工程问题中几何结构随时间演化的方法，例如。呈液晶状和液滴状。设计用于建模和模拟生物膜的新方法，使用表面有限元方法很好地近似完整曲率信息（即完整形状算子）。此外，我们将扩展生物膜表面有限元技术，将高维数据非线性降维到低维空间（用于数据分析和可视化）。研究的其他方面将为这里开发的方法创建开源软件，使用 PI 自己的软件包 FELICITY 和 AHF，以及其他开源选项（例如 Firedrake）。此外，PI 将利用他的“与科学家坐在一起”计划（如上所述）来教育中小学生。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。