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

这笔拨款将支持理解和操纵工程大人口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、超材料结构和车辆交通。一个关键的挑战是赋予诸如机器人群这样的系统在不同的集体行为之间自主切换的能力，例如，响应外部刺激。在其他系统中，如混合自主-手动交通，我们希望推动或激励智能体走向更理想的集体状态，例如，减少拥堵。系统时空集体行为的质变是在物理相变的框架下研究的。本研究将这些方法扩展到受控的大种群多智能体系统，并创建一个统一的框架来理解和触发这些系统中的相变。本研究旨在了解源自平均场对策和平均场控制框架的非标准动力系统的分岔和全局相空间结构。大种群多智能体系统的平均场控制理论结合了统计物理和最优控制的思想，并对大量相互作用的智能体在合作或非合作环境中表现最佳的场景进行了建模。由此产生的动力系统由时间上完全耦合的前向后非线性偏微分方程组成，它们的复杂性迄今为止阻碍了对解的性质的定性理解。被控集体行为的相变是闭环平均场控制问题解分叉的结果，因为问题参数如代价函数、惩罚和监督控制动作等是不同的。本研究将采用现有的描述性（前向）模型的局部分岔理论，如非线性薛定谔方程和群集，来表征平均场博弈和平均场控制理论的规定性或闭环（前向向后）模型的分岔。研究还将产生这些无限维系统的低阶模型。这种模型的相空间几何将使发现全局分岔和连接轨道负责在不同的集体行为之间切换。使用相空间几何来理解和诱导临界是迈向大规模受控动力系统的“热力学理论”的踏脚石。从这个项目中获得的见解可以证明在合理设计控制惩罚和/或激励以塑造各种应用中非线性代理的集体行为方面是有用的。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.