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来研究相互作用的“代理”或“粒子”的大型系统。主要目标是通过用平均场概念替换精确交互来降低具有大量代理的系统的复杂性。为了证明这种近似的合理性，现有理论通常假设代理是相同的或至少是无法区分的。在考虑物理学中的经典应用（例如静电势下的电子动力学）时，这种假设可能是有效的。研究人员将引入新的方法来消除无法区分的粒子或代理的假设，同时仍然涉及现实的奇异相互作用：这将允许考虑更复杂的应用，例如物理学中的多物种模型和生物神经元网络动力学模型。该项目为研究生提供培训研究机会。从数学上来说，不同的粒子或试剂会改变系统的结构。首先，它使它们成为非对称且不可交换的，这在尝试定义所谓的可观测量（例如子系统的联合定律）时会产生深远的影响。它还打破了一些抵消效应，例如相互作用的反对称性，许多当前方法都依赖这些抵消效应来处理奇异相互作用。该项目引入了绕过这些问题的新方法，例如新的二元性表述和扩展的边际概念。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和更广泛的影响审查标准进行评估，被认为值得支持。