Mathematical Aspects of Materials Science and Prediction with Expert Advice

这个项目有两个重点。第一个主题是数学和材料科学的交叉领域，涉及一类被称为“麦克斯韦晶格”的人工材料。这项工作将研究麦克斯韦晶格的宏观力学性质是如何由其微观结构决定的。提高对微观结构和宏观行为之间关系的理解有望促进更好的人工材料的设计。该项目的第二个重点在于数学和数据科学之间的接口，因此与国家科学基金会的十大理念之一（“利用数据革命”）保持一致。这项工作将集中在顺序决策的问题上，其中增量到达的数据必须组装成一个连贯的整体，尽管不一致；具体的重点将是最近引入的低秩矩阵补全方法。在这两个领域，该项目都将涉及博士生。参与这项研究的初级科学家将获得数学和关键应用领域的经验和广度。该项目的第一个重点是探索二维“麦克斯韦晶格”的非线性力学。根据定义，这些是周期晶格结构，其中遇到节点的平均边数为4；著名的Kagome晶格就是一个很好的例子。麦克斯韦格很有趣，因为在大多数情况下，它们是弹性简并的。关于它们的线弹性行为已经有了大量的文献；然而，这些结构的非线性力学仍然知之甚少，即使是很少或没有应变的变形。该项目的第一个推力将使用非线性均匀化和变分法的方法来探索麦克斯韦晶格的非线性力学。该项目的第二个重点是探索最近引入的低阶矩阵完井方法。这种方法依赖于一个两人、零和、顺序游戏，类似于被称为“专家建议预测”的在线机器学习模型。首席研究员最近使用了最优控制的方法来获得关于专家建议的预测的新见解；预计基于控制的方法也将对矩阵补全有用。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。

This project has two main thrusts. The first is a topic at the interface between mathematics and materials science, involving a class of artificial materials known as "Maxwell lattices." This work will study how the macroscopic mechanical properties of a Maxwell lattice are determined by its microscopic structure. An improved understanding of the relationship between microstructure and macroscopic behavior is expected to facilitate the design of better artificial materials. The project's second thrust lies at the interface between mathematics and data science, and is thus aligned with one of NSF's ten Big Ideas ("Harnessing the Data Revolution"). This work will focus on a problem of sequential decision making, in which data that arrive incrementally must be assembled into a coherent whole despite inconsistencies; the specific focus will be a recently-introduced approach to low-rank matrix completion. In both areas, the project will involve PhD students. The junior scientists involved in this research will gain experience and breadth in both mathematics and a key application area.The project's first thrust will explore the nonlinear mechanics of two-dimensional "Maxwell lattices." These are, by definition, periodic lattice structures in which the average number of edges meeting a node is four; the well-known Kagome lattice is a favorite example. Maxwell lattices are interesting because they are, in most cases, elastically degenerate. A large literature has developed concerning their linearly elastic behavior; however the nonlinear mechanics of these structures is still poorly understood, even for deformations with little or no strain. The project's first thrust will use methods from nonlinear homogenization and the calculus of variations to explore the nonlinear mechanics of Maxwell lattices. The project's second thrust will explore a recently introduced approach to low-rank matrix completion. This approach relies on a two-person, zero-sum, sequential game analogous to the model of online machine learning known as "prediction with expert advice." The principal investigator recently used methods from optimal control to achieve fresh insight concerning prediction with expert advice; it is expected that control-based methods will also be useful for matrix completion.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.