Data-Driven Discovery of Dynamics in Interacting Agent Systems and Linear Diffusion Processes

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The goal of this project is to develop data-driven methods for dynamical systems and specifically on interacting agent/particle systems and linear diffusion processes that arise in various disciplines such as opinion dynamics under social influence, prey-predator systems, flocking and swarming of animal groups, rumor/threat propagations over networks, and traffic flow over road networks. The project will focus on ideas from statistical learning for the discovery of governing laws and turning the observational data into equations that can be used for predictions. While machine learning techniques are particularly promising for this task their application to learning dynamical systems is still in its infancy. This project will develop efficient algorithms to learn unknown structures and parameters of the systems from various types of observational trajectory data, together with a rigorous quantitative framework to guide the selection of models that generalize well on unseen data. Students will be involved and trained in interdisciplinary aspects. The first part of the project addresses regression-based learning approaches to discover interaction laws between agents from various types of trajectory data, with applications to systems arising from physics, biology, ecology, and social sciences, using methods at the interface of machine learning and inverse problems. Systematic learning theories will be developed to study the well-posedness and model selections to achieve statistically optimal performance. The second part of the project will develop robust methods to recover linear diffusion processes over graphs from partial observations of evolving states, with applications to graph signal processing. In particular the project will develop sampling theorems to collect space-time samples as well as robust reconstruction algorithms. The sampling theorems will shed light on how to utilize dynamics over graphs and the structure of graphs to compensate for the loss of spatial information. Theoretical and algorithmic ramifications of the effects caused by imperfect data will be studied to test the proposed algorithms on synthetic and real data sets over a wide variety of graphs.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.