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.