Collaborative Research: Dynamics of Densities: Modeling, Control and Estimation

当今制导和控制方面的进步，从航空航天应用到现代设备技术，都需要更复杂的方法来处理海量数据和处理不确定性的概率性质。拟议的研究旨在开发新的计算技术和新的数学，以解决不确定系统的决策和控制问题，并以统一的方式调节动态系统的集合。典型的例子包括动力系统的集合，如无人区域车辆的集合、粒子束、受热波动的微机械系统、激光驱动的分子反应，或者网络上的资源流动的调度和控制。多个工程和科学学科将从该项目的成功完成中获益。建议框架的基础是新发现的关于概率密度的最优质量传输(OMT)的黎曼结构。随之而来的几何学为随机微分方程的分析提供了新的技术，为粒子的集体动力学提供了一个新的视角，导致了新的建模原则，并导致了设计适当的控制律的统一方法，以确保动力系统集合的稳定性和性能。密度函数成为主要的研究对象，因为它们代表了动力系统集合的状态。数据提供经验快照、统计数据和直方图，被视为密度函数的黎曼流形上的点。通过密度函数描述系统状态和不确定性与偏微分方程式中的新数学概念和工具是兼容的，这些概念和工具现在可以用于控制目的。这一新观点超越了控制应用，并可能影响非平衡统计力学、医学成像、过滤和信号分析。我们将开发的数学旨在扩展当前的OMT框架，以解决多维对象和张量场，从而在扩散张量成像、相互作用的动力学粒子的随机驱动密度流以及网络传输应用中的应用将成为研究的重点。该奖项反映了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。

Present day advances in guidance and control, from aerospace applications to modern device technologies, require ever more sophisticated ways to deal with the deluge of data and to cope with the probabilistic nature of uncertainty. The proposed research aims at developing novel computational techniques and new mathematics that address problems in decision and control of uncertain systems and in regulating collections of dynamical systems in a unified manner. Typical examples include the steering of collections of dynamical systems such as a collection of unmanned area vehicles, a particle beam, micro-mechanical systems subject to thermal fluctuations, laser-driven molecular reactions, or the scheduling and control of the flow of resources across a network. Multiple engineering and scientific disciplines stand to gain from successful completion of the project.The basis of the proposed framework is the newly discovered Riemannian structure of Optimal Mass Transport (OMT) on probability densities. The ensuing geometry allows new techniques for the analysis of stochastic differential equations, a new perspective of collective dynamics of particles that leads to new modeling principles, and leads to a unified approach for designing suitable control laws that ensure stability and performance of collections of dynamical systems. Density functions become the primary object of study as these represent the state of the collection of dynamical systems. Data provide empirical snapshots, statistics, and histograms viewed as points on the Riemannian manifold of density functions. The description of system-states and uncertainty via density functions is compatible with novel mathematical concepts and tools in partial differential equations that can be now used for control purposes. This new viewpoint transcends control applications and may impact non-equilibrium statistical mechanics, medical imaging, filtering and signal analysis. The mathematics that we will develop aim to extend the current framework OMT to address multi-dimensional objects and tensor fields and, thereby, applications in diffusion tensor imaging, stochastically driven density flows of interacting dynamical particles, and network transport applications will be a focus of the research.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.