Phase Space Geometry of Critical Transitions in Collective Behavior Modeled by Mean Field Type Control Problems

平均场类型控制问题建模的集体行为关键转变的相空间几何

• 批准号: 2102112
• 负责人: Piyush Grover
• 金额: $ 31.15万
• 依托单位: University of Nebraska-Lincoln
• 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: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2102112&HistoricalAwards=false
• 关键词: Phase; Space; Geometry; Critical; Transitions

This grant will support research on understanding and manipulating the collective behavior of engineered large-population multi-agent systems, promoting both the progress of science and advancing national prosperity. Examples of such systems include mobile robot swarms, smart grid, metamaterial structures, and vehicular traffic. A key challenge is to impart systems such as robot swarms with the capability to autonomously switch between different collective behaviors, e.g., in response to external stimuli. In other systems such as mixed autonomous-manual traffic, it is desirable to nudge or incentivize the agents toward more desirable collective state(s), e.g., to reduce congestion. Qualitative changes in spatiotemporal collective behavior of a system are studied under the umbrella of phase transitions in physics. This research will extend such methods to controlled large-population multi-agent systems, and create a unifying framework for understanding and triggering phase transitions in such systems. This research aims to understand bifurcations and global phase space structure of non-standard dynamical systems originating in the mean field games and mean field control framework. The mean field control theory for large-population multi-agent systems combines ideas from statistical physics with optimal control, and models scenarios where a large number of interacting agents are acting optimally, either in cooperative or non-cooperative setting. The resulting dynamical systems consist of fully-coupled forward-backward in time nonlinear partial differential equations, and their complexity has to date prevented qualitative understanding of the nature of solutions. Phase transitions in the controlled collective behavior are the result of bifurcations of the solutions of closed-loop mean field control problems as problem parameters, such as cost functions, penalties, and supervisory control action, etc., are varied. This research will adapt local bifurcation theory of existing descriptive (forward) models such as the nonlinear Schrodinger equations and flocking to characterize bifurcations in prescriptive or closed-loop (forward-backward) models of mean field games and mean field control theory. The research will also produce low-order models of these infinite dimensional systems. The phase space geometry of such models will enable discovery of global bifurcations and connecting orbits responsible for switching between different collective behaviors. The use of phase space geometry to understand and induce criticality is a stepping stone towards a ‘theory of thermodynamics’ of large-scale controlled dynamical systems. The insight gained from this project can prove useful in rational design of control penalties and/or incentives to shape the collective behavior of nonlinear agents for diverse applications.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.

这笔赠款将支持理解和操纵工程化的大规模多智能体系统的集体行为的研究，促进科学进步和促进国家繁荣，此类系统的例子包括移动机器人群、智能电网、超材料结构和车辆。一个关键的挑战是让机器人群等系统能够在不同的集体行为之间自主切换，例如响应外部刺激。推动或激励智能体走向更理想的集体状态，例如，减少拥塞。在物理学相变的框架下研究系统时空集体行为的定性变化。这项研究旨在了解源自平均场博弈和非标准动力系统的分岔和全局相空间结构。大规模多智能体系统的平均场控制理论将统计物理学的思想与最优控制相结合，并对大量交互智能体在合作或非合作环境中表现最佳的场景进行建模。由此产生的动力系统由完全耦合的前向-后向时间非线性偏微分方程组成，其复杂性迄今为止阻碍了对解的性质的定性理解受控集体行为中的相变是解的分岔的结果。闭环均值场控制问题作为问题参数，例如成本函数、惩罚和监督控制动作等，是多种多样的。本研究将采用现有描述性（正向）模型的局部分岔理论，例如非线性薛定谔方程和聚集来表征分岔。该研究还将产生这些无限维系统的低阶模型，这些模型的相空间几何将使全局的发现成为可能。分叉和连接轨道负责在不同的集体行为之间进行切换，使用相空间几何来理解和诱导临界性是迈向大规模受控动力系统“热力学理论”的垫脚石，从这个项目中获得的见解可以证明。有助于合理设计控制惩罚和/或激励措施，以塑造不同应用的非线性代理的集体行为。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。