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

优化、采样和过滤出现在广泛的领域，从机器学习、神经网络、逆问题、数据同化和模型参数估计到更经典的领域，如经济学、计算生物学、数学金融和统计物理。该项目将有助于设计和分析可实现的和计算效率高的算法，从相互作用的粒子系统的角度进行优化，采样和过滤。它直接解决的实际问题，对于一个给定的误差阈值和计算预算，如何选择算法，其参数，其初始条件，使一个获得所需的精度输出。作为该项目的一部分开发的方法将用于建模细胞骨架网络，这是一个具有挑战性的生物工程问题，不仅有助于阐明细胞过程，而且还可以用于开发可编程活性物质设备。该项目还包括多方面的教育和推广计划，包括研究生和本科生的研究监督，课程开发，以及两个关于数据科学和应用数学教育的研讨会，重点是负责任地使用数据和人工智能，以及如何在低成本环境下实现高质量的数据科学和应用数学教育。这个项目的首要目标是建立一个统一的理论，为一大类无衍生物的优化，采样和过滤算法使用离散到连续连接。无导数方法在使用黑盒程序的应用程序设置中特别重要，或者在梯度太昂贵而无法获得的情况下。该项目的核心是从交互粒子系统的角度重新制定这些优化，采样和过滤算法的策略，将它们整合到一个新的统一的数学框架中。这种观点提供了利用偏微分方程的工具来分析与驱动这些算法的相互作用粒子相关的概率的方法。该项目分为三个不同但相互关联的研究方向：（1）基于共识的优化和采样方法，（2）用于逆问题和滤波的集合卡尔曼方法，以及（3）细胞骨架网络中的自组织。这些方法处于模型驱动和数据驱动方法的界面。拟议的工作坐落在偏微分方程理论，混沌传播，随机分析，动力学理论，优化，数据同化，数学建模和生物-该奖项反映了NSF的法定使命，并通过使用基金会的智力价值进行评估，更广泛的影响审查标准。

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.