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

职业：相互作用的粒子系统及其平均场偏微分方程：当非线性模型遇到数据时

• 批准号: 2340762
• 负责人: Franca Hoffmann
• 金额: $ 50万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2029-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2340762&HistoricalAwards=false
• 关键词: CAREER; Interacting; Particle; Systems; their

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.

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古典的领域，例如经济学，计算生物学，数学金融和统计物理学等更经典领域，优化，采样和过滤出现在广泛的领域中。该项目将有助于从相互作用的粒子系统的角度从可实施和计算有效算法进行优化，采样和过滤的设计和分析。它直接解决了如何在给定的误差阈值和计算预算中选择算法，其参数及其初始条件，从而以所需的准确性获得输出。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，它不仅有助于阐明蜂窝过程，而且还可以对开发可编程的活动物质设备有用。该项目还纳入了多方面的教育和外展计划，包括研究生和学生研究监督，课程开发以及两个有关数据科学和应用数学教育的研讨会，这些教育和应用数学教育的重点是负责任的数据和AI，以及如何实现高质量数据科学的高质量数据科学教育，以在低水平的环境中进行庞大的数学范围。使用离散到核电连接采样和过滤算法。在使用黑框程序或梯度太高以至于无法获得的地方，无衍生的方法在应用程序设置中尤为重要。该项目的核心是从相互作用的粒子系统的角度重新调整这些优化，采样和过滤算法的策略，将它们集成到一个新的统一数学框架中。该观点为从部分微分方程中利用工具的方式提供了方法，以分析与驱动这些算法的相互作用粒子相关的概率。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）用于反问题和过滤的集合卡尔曼方法，以及（3）细胞骨架网络中的自组织。这些方法在于模型驱动和数据驱动方法的界面。所提出的工作是部分差异方程式，混乱的传播，随机分析，动态理论，动力学理论，优化，数据同化，数学建模和生物学工程的固有方法的转移以及对所有这些奖项之间的新贡献的固有方法转移。影响审查标准。