CAREER: Interacting Particle Systems and their Mean-Field PDEs: when nonlinear models meet data

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Optimization, sampling and filtering appear across a wide range of fields, from machine learning, neural networks, inverse problems, data assimilation and model parameter estimation to more classical areas such as economics, computational biology, mathematical finance and statistical physics. This project will contribute to the design and analysis of implementable and computationally efficient algorithms for optimization, sampling and filtering from the perspective of interacting particle systems. It directly addresses the practical matter of how, for a given error threshold and computational budget, to choose the algorithm, its parameters, and its initial conditions such that one obtains an output at a desired accuracy. The methodologies developed as part of this project will be used for modeling cytoskeletal networks, a challenging bio-engineering problem, that will not only help shed light on cellular processes but could also be useful in developing programmable active matter devices. This project also incorporates multi-faceted education and outreach plans, including graduate and undergraduate student research supervisions, course development, and two workshops on data science and applied mathematics education which are focused on responsible use of data and AI as well as how to achieve high-quality data science and applied mathematics education in low-resource environments.The overarching goal of this project is to build a unified theory for a large class of derivative-free optimization, sampling and filtering algorithms using discrete-to-continuum connections. Derivative-free approaches are particularly important in application settings using black-box procedures, or where gradients are too costly to obtain. Central to this project is the strategy to reformulate these optimization, sampling and filtering algorithms from the perspective of interacting particle systems, integrating them into a new, unified mathematical framework. This perspective provides ways for leveraging tools from partial differential equations for the analysis of the probabilities associated with the interacting particles driving these algorithms. The project is divided into three different, but interrelated, research directions: (1) consensus-based approaches for optimization and sampling, (2) ensemble Kalman methods for inverse problems and filtering, and (3) self-organization in cytoskeletal networks. These methodologies lie at the interface of model-driven and data-driven approaches. The proposed work sits at the intersection of the theory of partial differential equations, propagation of chaos, stochastic analysis, kinetic theory, optimization, data assimilation, mathematical modeling and bio-engineering with inherent methodology transfer between these fields and new contributions to all of them.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.