FRG: Collaborative Research: Non-Smooth Geometry, Spectral Theory, and Data: Learning and Representing Projections of Complex Systems

FRG：协作研究：非光滑几何、谱理论和数据：学习和表示复杂系统的投影

基本信息

批准号：
1854299
负责人：
John Harlim
金额：
$ 34.34万
依托单位：
Pennsylvania State Univ University Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854299&HistoricalAwards=false
关键词：
FRG Collaborative Research Non Smooth

项目摘要

Complex, time-evolving systems are ubiquitous in nature and society, with examples ranging from the Earth's weather and climate, to the function and dynamics of biomolecules, and the behavior of markets and economies. Despite their apparent complexity, many such systems exhibit a form of underlying organized structure (``building blocks''), whose discovery would enhance our ability to understand and predict a wide range of phenomena. The goal of this project is to develop the next generation of mathematical and algorithmic tools that can harness the information content of large datasets acquired from experiments and observations to create coherent representations of complex systems, and use these representations to perform prediction, and ultimately, control. These objectives will be addressed through a novel combination of mathematical techniques, bridging dynamical systems theory and differential geometry with machine learning and data science. The newly developed techniques will be tested and applied in real-world problems through collaboration with domain experts in the areas of climate dynamics, space physics, and condensed matter physics. The project will also contribute to STEM workforce and curricular development through training of students and postdoctoral researchers, and design of multi-disciplinary lecture courses. In particular, this project will support one graduate student at each of the three universities involved.The modern scientific method is undergoing an evolutionary change wherein large data sets and machine learning algorithms have the potential to outperform classical first-principles approaches for certain complex phenomena. For these tools to be accepted by the scientific community, a rigorous mathematical framework is required to match the verifiability and quantifiability of the classical modeling approach. Recently, a new tool called the diffusion forecast has been developed based on provably consistent estimators, which learn the unknown structure of a large class of stochastic dynamical systems on manifolds. Moreover, the results of many published numerical experiments indicate that this framework can be applied far beyond the restricted context of the current theory. In particular, the evidence suggests that the consistency proofs can be extended to non-autonomous projections of complex systems, deterministic chaotic systems represented by non-compact operators, non-smooth domains such as fractal attractors, and even generalized tensors on metric-measure spaces. This project will undertake a rigorous mathematical unification of these problems, leading to transformative advances in our ability to model and describe complex systems.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.

复杂的，随着时间的推移系统在自然和社会上无处不在，范围从地球的天气和气候到生物分子的功能和动态以及市场和经济的行为。尽管它们明显复杂，但许多这样的系统表现出一种基本的有组织结构（````构建''''）的形式，它们的发现将增强我们理解和预测广泛现象的能力。该项目的目的是开发下一代的数学和算法工具，这些工具可以利用从实验和观察值中获取的大型数据集的信息内容，以创建复杂系统的连贯表示，并使用这些表示形式执行预测，并最终控制控制。这些目标将通过数学技术，桥接动力学系统理论和机器学习和数据科学的差异几何形状的新颖组合来解决。新开发的技术将通过与气候动态，空间物理和凝结物理物理学领域的领域专家的合作进行测试和应用。该项目还将通过培训学生和博士后研究人员以及多学科讲座课程的设计来为STEM劳动力和课程发展做出贡献。特别是，该项目将在所涉及的三所大学中的每所大学中支持一名研究生。现代科学方法正在经历进化变化，其中大型数据集和机器学习算法有可能超过经典的第一原理方法来实现某些复杂现象。要使这些工具被科学界接受，需要一个严格的数学框架来匹配经典建模方法的可验证性和量化性。最近，基于可证明的一致估计器开发了一种称为“扩散预测”的新工具，该估计量学习了歧管上一类随机动力学系统的未知结构。此外，许多已发表的数值实验的结果表明，该框架可以远远超出当前理论的受限背景。特别是，证据表明，一致性证明可以扩展到复杂系统的非自主投影，由非紧凑型操作员，非平滑域（例如分形吸引子）所代表的确定性混沌系统，甚至是公认的张量。该项目将对这些问题进行严格的数学统一，从而导致我们建模和描述复杂系统的能力的变革性进步。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的评估评估标准来通过评估来支持的。