Defense Advanced Research Projects AgencyTagged Content List

Mathematics

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The goal of the Fundamental Design (FUN Design) program is to determine whether we can develop or discover a new set of building blocks to describe conceptual designs. The design building blocks will capture the components’ underlying physics allowing a family of nonintuitive solutions to be generated.
Machine learning has shown remarkable success across many application areas in recent years, leveraging advances in computing power and the availability of large sets of training data. It provides a tremendous opportunity to deploy data-driven systems in more complex and interactive tasks including personalized autonomy, agile robotics, self-driving vehicles, and smart cities. Despite dramatic progress, the machine learning community still lacks an understanding of the trade-offs and mathematical limitations of related technologies for a given domain, problem, or dataset.
The science of network analysis is in its infancy. Currently, the structure of real-world networks is described only in terms of coarse and basic details such as diameter, degree distribution, etc. In addition, as networks become large, many problems are intractable as the classical algorithms for these problems run in exponential time with respect to the size of the graph. A large number of important problems (e.g., structural and functional brain dynamics or gene-protein and disease networks) can be formulated as graph problems. A comprehensive mathematical understanding of large networks is needed in order to effectively apply and scale graph-based network analysis techniques for use in DoD-relevant scenarios.
The social sciences can play important roles in assisting military planners and decision-makers who are trying to understand complex human social behaviors and systems, potentially facilitating a wide range of missions including humanitarian, stability, and counter-insurgency operations. Current social science approaches to studying behavior rely on a variety of modeling methods—both qualitative and quantitative—which seek to make inferences about the causes of social phenomena on the basis of observations in the real-world. Yet little is known about how accurate these methods and models really are, let alone whether the connections they observe and predict are truly matters of cause and effect or mere correlations.
The Lagrange program seeks to develop new mathematical approaches to optimization problems in uncertain, dynamic, multiscale, and high-dimensional settings. By bridging methodologies developed for both discrete and continuous optimizations, Lagrange aims to enable solutions for complex, realistic problems that involve dynamic environments, rapidly changing requirements, and increasing or decreasing amounts of information.