When Does a Diverse Initial Solution Set Lead to Better Engineering Design Outcomes?

设计高性能的工程系统--例如，燃油效率高的飞机、医疗设备、新的制造和农业设备--需要在许多可能的选择中寻找高质量的解决方案。对于复杂的问题，可能很难找到好的解决方案，人们使用的一条经验法则是首先从一系列不同的选项开始。人们的直觉是，拥有不同的起点将使人更有可能最终找到一个好的最终解决方案。但情况一直都是这样吗？对于哪些类型的问题，拥有不同的初始解决方案有助于还是阻碍创建更好的最终解决方案？该奖项支持对数学模型的基础研究，以回答这些问题。具体地说，这项研究研究了新的模型，这些模型1)绘制出工程问题在结构和人们用来解决它们的过程方面的不同之处，2)预测每种类型的工程问题的不同初始解决方案是否会导致更高质量的最终解决方案。这项工作将通过提供一种降低成本的方法来影响社会，并提高多个行业的工程师解决问题的能力。这个项目的技术目标是(1)通过使用范畴理论和不同类型的解空间之间的同构来统一工程系统的各种表示；(2)使用随机矩阵理论来编码搜索和设计过程以跟踪该统一表示上的解；(3)使用子模块顶点覆盖的度量来计算初始和最终解集合的多样性；(4)使用计算模拟和人类参与者的行为实验来测试该模型；(5)利用矩阵浓度不等式研究了模型的有限样本行为。这个项目的结果将通过提供一个可测试的模型来推动工程学领域的发展，提供一个可测试的模型，说明不同的初始解决方案如何影响从人类决策和团队到计算优化算法的各种任务。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Designing high-performing engineering systems--for example, fuel-efficient aircraft, medical devices, new manufacturing and agricultural equipment--requires searching for high-quality solutions among many possible options. It can be difficult to find good solutions to complex problems, and one rule-of-thumb people use is to first start with a diverse set of options. The intuition is that having diverse starting points will make it more likely that one will eventually find a good final solution. But is this always the case? For what types of problems does having diverse initial solutions help versus hinder creating better final solutions? This award supports fundamental research into mathematical models that answer these questions. Specifically, this research studies new models that 1) map how engineering problems differ from each other in terms of their structure and the processes people use to solve them, and 2) predict whether diverse initial solutions across each type of engineering problem lead to a higher quality final solution. The work will impact society by providing a way to reduce costs and improve problem-solving performance of engineers across multiple industries. The mathematical models will also provide foundational understanding to other fields where similar questions arise, such as in biology, computer science, and physics.The technical objectives of this project are to (1) unify various representations of engineering systems through the use of Category Theory and isomorphisms between types of solution spaces; (2) encode search and design processes using Random Matrix Theory to track solutions over that unified representation; (3) use metrics from Submodular Vertex Coverage to calculate the diversity of initial and final solution sets; (4) test the model across five benchmark problems of increasing difficulty and complexity, using computational simulations and behavioral experiments with human participants; and (5) study the finite sample behavior of the model using Matrix Concentration Inequalities. The results from this project will advance the field of engineering by providing a testable model for how diverse initial solutions impact a wide range of tasks from human decision making and teaming to computational optimization algorithms.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.