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

复杂的、随时间变化的系统在自然界和社会中无处不在，从地球的天气和气候，到生物分子的功能和动力学，再到市场和经济的行为，都是这样的例子。尽管它们看起来很复杂，但许多这样的系统显示出一种潜在的有组织的结构形式（“构建块”），它们的发现将增强我们理解和预测各种现象的能力。该项目的目标是开发下一代数学和算法工具，这些工具可以利用从实验和观察中获得的大型数据集的信息内容来创建复杂系统的连贯表示，并使用这些表示来进行预测，并最终实现控制。这些目标将通过数学技术的新颖组合来解决，将动力系统理论和微分几何与机器学习和数据科学联系起来。新开发的技术将通过与气候动力学、空间物理和凝聚态物理领域的专家合作，在现实世界的问题中进行测试和应用。该项目还将通过培训学生和博士后研究人员，以及设计多学科讲座课程，为STEM劳动力和课程发展做出贡献。现代科学方法正在经历一场进化变革，其中大型数据集和机器学习算法有可能在某些复杂现象中超越经典的第一原理方法。为了使这些工具被科学界所接受，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可量化性。最近，一种基于可证明相合估计量的扩散预测工具被开发出来，它学习了流形上一大类随机动力系统的未知结构。此外，许多已发表的数值实验结果表明，该框架可以远远超出当前理论的限制范围。特别是，证据表明一致性证明可以推广到复杂系统的非自治投影、由非紧算子表示的确定性混沌系统、分形吸引子等非光滑域，甚至度量-度量空间上的广义张量。该项目将对这些问题进行严格的数学统一，从而在我们建模和描述复杂系统的能力方面取得革命性的进步。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。

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. 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.