Multiscale Effects and Tail Events for Infinite-Dimensional Processes and Interacting Particle Systems

无限维过程和相互作用粒子系统的多尺度效应和尾部事件

• 批准号: 2107856
• 负责人: Konstantinos Spiliopoulos
• 金额: $ 25.58万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2107856&HistoricalAwards=false
• 关键词: Multiscale; Effects; Tail; Events; Infinite

Probabilistic models are commonly used to represent physical, biological, and financial phenomena that are often too complex to solve, approximate or even simulate on computers. One of the challenges facing applied mathematics and probability is to obtain accurate and provably efficient methods to approximate and simulate a range of complex systems using probabilistic models. The primary purpose of this research is to rigorously investigate problems related to multiscale systems, rare events, and related simulation methods. The principal investigator (PI) is interested in studying stochastic dynamical systems that may have different time scales and quantifying related events that may be rare on a given time scale but can have important consequences for the system itself. The questions of interest in this project are motivated both by fundamental mathematical questions and by a broad array of questions in other branches of science. Examples range from estimation of rare event probabilities in chemical physics, hydrodynamics, coupled chemical reactions with spatially-dependent diffusion, to population genetics and opinion dynamics. This research project is integrated with an educational program that is designed to help in the training of undergraduate and graduate students in applied mathematics, physics, engineering, and chemistry in the exploration of rare events, multiscale processes and their analysis and simulation.The PI is interested in the large deviations regime (tail events) as well as in the moderate deviations regime (the gap between the typical center and the tail of the distribution). Due to the lack of explicit solutions, one has to rely on approximation and simulation methods and for this reason rigorous development of provably efficient approximation methods and simulation Monte Carlo methods is essential. In a closely related direction, the PI develops a rigorous theory of metastability for infinite dimensional dynamical systems that may have multiple scales, interacting with the rare events of interest. Moreover, the PI is interested in the effect of multiple scales on tail events associated to interacting particle systems. Systems of interacting diffusions arise in many areas of science, theory of random matrices, mathematical biology, neural networks in machine learning and optimization, construction of Kahler-Einstein metrics, opinion dynamics, finance, and engineering. The proposed work leads to the development of a rigorous mathematical framework that allows to design provably efficient algorithms associated to modeling of rare events. It will crystalize concepts and methods that are not well understood such as the effect of metastability on Monte Carlo methods and of multiple scales on large deviations and Monte Carlo methods for stochastic dynamical systems in infinite dimensions and for interacting particle systems.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.

概率模型通常用于表示物理、生物和金融现象，这些现象往往过于复杂，无法在计算机上求解、近似甚至模拟。应用数学和概率学面临的挑战之一是利用概率模型获得精确和被证明有效的方法来近似和模拟一系列复杂系统。这项研究的主要目的是严格调查与多尺度系统、罕见事件和相关模拟方法有关的问题。首席研究员(PI)感兴趣的是研究可能具有不同时间尺度的随机动力系统，并量化在给定时间尺度上可能罕见但可能对系统本身产生重要后果的相关事件。这个项目中感兴趣的问题既受到基本数学问题的推动，也受到其他科学分支的广泛问题的推动。例子包括从化学物理、流体动力学、与空间相关扩散的耦合化学反应到群体遗传学和意见动力学中的罕见事件概率的估计。这项研究项目与一个教育项目相结合，该项目旨在帮助应用数学、物理、工程和化学专业的本科生和研究生培养探索罕见事件、多尺度过程及其分析和模拟的能力。PI对大偏差区域(尾部事件)和中等偏差区域(典型分布中心和尾部之间的差距)感兴趣。由于缺乏显式解，人们不得不依赖于近似和模拟方法，因此严格开发可证明有效的近似方法和模拟蒙特卡罗方法是必不可少的。在一个密切相关的方向上，PI为可能具有多个尺度的无限维动力系统发展了一种严格的亚稳性理论，该系统与感兴趣的罕见事件相互作用。此外，PI感兴趣的是多个尺度对与相互作用的粒子系统相关联的尾部事件的影响。交互扩散系统出现在科学、随机矩阵理论、数学生物学、机器学习和优化中的神经网络、卡勒-爱因斯坦度量的构建、观点动力学、金融和工程的许多领域。这项拟议的工作导致了一种严格的数学框架的发展，该框架允许设计与罕见事件建模相关的经证明有效的算法。它将明确那些还不是很好理解的概念和方法，例如亚稳性对蒙特卡罗方法和大偏差上多尺度的影响，以及无限维随机动力系统和相互作用粒子系统的蒙特卡罗方法。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

PDE-constrained Models with Neural Network Terms: Optimization and Global Convergence

• DOI: 10.1016/j.jcp.2023.112016
• 发表时间: 2021-05
• 期刊: ArXiv
• 作者: Justin A. Sirignano;J. MacArt;K. Spiliopoulos

Large deviations for interacting multiscale particle systems

• DOI: 10.1016/j.spa.2022.09.010
• 发表时间: 2020-11
• 期刊: Stochastic Processes and their Applications
• 影响因子: 1.4
• 作者: Zachary Bezemek;K. Spiliopoulos

Moderate deviations for systems of slow–fast stochastic reaction–diffusion equations

• DOI: 10.1007/s40072-022-00236-y
• 发表时间: 2020-12
• 期刊: Stochastics and Partial Differential Equations: Analysis and Computations
• 作者: Ioannis Gasteratos;M. Salins;K. Spiliopoulos

Normalization effects on deep neural networks

归一化对深度神经网络的影响

• DOI: 10.3934/fods.2023004
• 发表时间: 2023
• 期刊: Foundations of Data Science
• 影响因子: 2.3
• 作者: Yu, Jiahui;Spiliopoulos, Konstantinos
• 通讯作者: Spiliopoulos, Konstantinos

Online Adjoint Methods for Optimization of PDEs

偏微分方程优化的在线伴随方法

• DOI: 10.1007/s00245-022-09852-5
• 发表时间: 2022
• 期刊: Applied Mathematics & Optimization
• 影响因子: 1.8
• 作者: Sirignano, Justin;Spiliopoulos, Konstantinos
• 通讯作者: Spiliopoulos, Konstantinos