Charting a New Paradigm for Large Non-Exchangeable Multi-Agent and Many-Particle Systems

为大型不可交换多代理和多粒子系统绘制新范式

基本信息

批准号：
2205694
负责人：
Pierre-Emmanuel Jabin
金额：
$ 30万
依托单位：
Pennsylvania State Univ University Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-15 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2205694&HistoricalAwards=false
关键词：
Charting New Paradigm Large Non

项目摘要

The project will develop a new framework for the study of large systems of interacting "agents" or "particles". The main goal is to obtain a reduction in complexity for systems with a very large number of agents, by replacing exact interactions with a notion of mean field. Existing theories, to justify this approximation, typically assume that the agents are identical or at least indistinguishable. This assumption can be valid when considering classical applications in Physics, such as the dynamics of electrons under an electrostatic potential. The investigator will introduce new approaches to remove the assumption of indistinguishable particles or agents while still involving realistic singular interactions: This will allow to consider more complex applications such as multi-species models in Physics and models for the dynamics of networks of biological neurons. The project provides training research opportunities for graduate students. Mathematically speaking having non-identical particles or agents changes the structure of the systems. First, it makes them non-symmetric and non-exchangeable, which has deep consequences when trying to define so-called observables, such as the joint law of a sub-system. It also breaks some of the cancellation effects, for example the anti-symmetry of the interaction, that many current methods relied on to handle singular interactions. The project introduces novel methods to bypass those issues such as new duality formulations and an extended notion of marginal.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.

该项目将开发一个新的框架，用于研究大型相互作用的“代理”或“颗粒”的系统。主要目标是通过用平均场的概念代替精确的相互作用来获得大量试剂的系统的复杂性降低。现有理论是为了证明这种近似值的合理性，通常假定代理是相同或至少无法区分的。当考虑物理中的经典应用，例如静电电位下的电子动力学时，此假设可能是有效的。研究者将引入新方法来消除无法区分的颗粒或代理的假设，同时仍涉及现实的奇异相互作用：这将允许考虑更复杂的应用，例如物理学中的多种物理模型和生物神经元网络的动力学。该项目为研究生提供了培训机会。从数学上讲，没有相同的颗粒或代理会改变系统的结构。首先，它使它们变得不对称且不可交换，这在试图定义所谓的可观察物（例如子系统的联合定律）时会产生深远的后果。它还打破了一些取消效应，例如相互作用的反对称性，许多当前方法都依赖于处理奇异相互作用。该项目介绍了新的方法，以绕过这些问题，例如新的二元表述和边缘的扩展概念。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来支持。