CAREER: Connecting Mathematical Models Across Scales

该职业奖支持旨在开发复杂系统的简单模型的理论和计算研究。复杂系统由大量较小的组件组成，这些组件以相互作用的方式产生有趣的集体行为，其中组件协同行动。这类系统的例子包括材料、生物途径、神经网络以及许多工程系统和社会网络。复杂的系统很难建模。通常，当数学模型足够简单，能够捕捉感兴趣现象的相关部分而忽略无关细节时，它们是最有用的。对于复杂的系统，很难从一开始就知道哪些组件是相关的，哪些是不相关的。因此，现实世界系统的模型往往过于复杂，难以处理，并且与更简洁的表示相比，预测能力有限。该项目利用了将信息论的数学领域与微分几何和拓扑学相结合的最新进展。其基本思想是利用一种系统的方法，从一个复杂系统的模型中剔除不相关的复杂性，直到得到一个足够简单的模型。通过列举所有产生的近似，科学家从物理系统的复杂、详细的表示，到各种类型的近似和它们所描述的系统行为，有了一个路线图。换句话说，这个过程充当了跨越尺度的数学桥梁：将微观机制与系统级现象联系起来。在这个项目中，PI将把这些新的数学和相关的计算工具应用于三个目标应用领域：合金的晶体结构、发育途径的生化动力学和神经元网络。通过更好地理解数学模型如何反映相关的物理细节，模型将更好地预测复杂系统的行为，并使复杂材料和过程的更复杂的设计和控制成为可能。这项工作还包括一个教育组成部分，将发展跨学科科学的教学法。这种教学法将以一系列研讨会的形式集中于大学教员；大学生以多系、专题的形式开设课程；并以高中教师教学工作坊和网络资源的形式为高中科学教师提供帮助。该职业奖支持旨在开发复杂系统的简单模型的理论和计算研究。简单的数学模型在科学探究中一直扮演着重要的角色。对于具有分离尺度或对称性的系统，有一些标准技术可以从复杂的机械描述中构造简洁的表示。将信息论与微分几何和拓扑学相结合的最新进展为复杂物理系统模型中机制和现象之间的关系提供了新的推理方法。在这项研究中，PI采用的方法是将多参数模型的预测形成嵌入在抽象数据空间中的流形。已经观察到，典型的模型流形是有界的，而简单的、近似的模型存在于模型流形的边界上。流形边界近似法构造了一系列极限近似，即对应于一系列简单近似模型的“小参数”。本研究将这些结果扩展到三个目标应用领域：合金晶体结构、发育途径的生化动力学和神经元网络。在这些目标应用领域中，PI将基于“最高模型”的概念开发一种新的建模方法，即同时解释几个行为的最小模型。当与流形边界近似方法相结合时，最高模型概念使人们能够从重叠系统组件的几个模型中引导到更大系统的模型，直接探索其中的计算挑战性或不可行的。这项工作还包括一个教育组成部分，将发展跨学科科学的教学法。这种教学法将以一系列研讨会的形式针对大学教员；大学生以多系、专题的形式开设课程；并以高中教师教学工作坊和网络资源的形式为高中科学教师提供帮助。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。

NONTECHNICAL SUMMARYThis CAREER award supports theoretical and computational research aimed at developing simple models of complex systems. Complex systems are made up of a large number of smaller components that interact in ways that can lead to interesting collective behaviors where the components act in concert. Examples of such systems include materials, biological pathways, and neural networks as well as many engineered systems and social networks. Complex systems are difficult to model. Usually, mathematical models are most useful when they are sufficiently simple to capture the relevant parts of the phenomenon of interest while ignoring irrelevant details. For complex systems, it is difficult to know from the start which components are relevant and which are irrelevant. Consequently, models of real-world systems tend to be overly complicated, difficult to work with, and have limited predictive power compared to more parsimonious representations.This project leverages recent advances that combine the mathematical areas of information theory with differential geometry and topology. The basic idea utilizes a systematic method for pruning irrelevant complications from a model of a complex system until a sufficiently simple model is obtained. By enumerating all of the resulting approximations, the scientist has a roadmap from the complicated, detailed representation of the physical system, through various types of approximations and the system behaviors they describe. Put another way, the process acts as a mathematical bridge across scales: connecting microscopic mechanisms to systems-level phenomena. In this project, the PI will apply these new mathematical and associated computational tools to three target application areas: crystal structures of alloys, biochemical kinetics of developmental pathways, and networks of neurons. By better understanding how mathematical models reflect relevant physical details, models will be better at predicting the behavior of complex systems and enable more sophisticated design and control of complicated materials and processes. This work also includes an educational component that will develop a pedagogy of interdisciplinary science. This pedagogy will focus on university faculty in the form of a seminar series; university students in the form of a multi-department, special topics course; and high school teachers in the form of teaching workshops and web resources for high school science teachers.TECHNICAL SUMMARY This CAREER award supports theoretical and computational research aimed at developing simple models of complex systems. Simple mathematical models have always played an important role in scientific inquiry. For systems with separated scales or symmetries, there are standard techniques for constructing parsimonious representations from complicated, mechanistic descriptions. Recent advances combining information theory and differential geometry and topology suggest new methods for reasoning about the relationship between mechanisms and phenomena in models of complex physical systems. For this research, the PI takes the approach that the predictions of a multi-parameter model form a manifold embedded in an abstract data space. It has been observed that typical model manifolds are bounded and that simple, approximate models reside on the boundary of the model manifold. The Manifold Boundary Approximation Method constructs a sequence of limiting approximations, that is, "small parameters" that corresponds to a sequence of simple, approximate models. This research will extend these results to three target application areas: alloy crystal structure, biochemical kinetics of developmental pathways, and networks of neurons. Within these target application areas, the PI will develop a new approach to modeling based on the concept of a "supremum model", that is, the minimal model that simultaneously explains several behaviors. When combined with the Manifold Boundary Approximation Method, the supremum model concept enables one to bootstrap from several models of overlapping system components to models of much larger systems, the direct exploration of which would be computationally challenging or infeasible.This work also includes an educational component that will develop a pedagogy of interdisciplinary science. This pedagogy will target university faculty in the form of a seminar series; university students in the form of a multi-department, special topics course; and high school teachers in the form of teaching workshops and web resources for high school science teachers.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.