AF: Small: Data-Driven Model Reduction for Optimal Control of Large-Scale Systems

动力系统是建模、预测和控制物理现象的主要工具，从复杂微电子设备的散热，到大型风力涡轮机的振动抑制，再到流动模拟。动态系统的最优控制在许多科学和工程应用中起着重要的作用，在这些应用中，人们希望产生输入来改善系统的性能。然而，在建模阶段不断增加的包括更多细节的需求不可避免地导致更大规模，更复杂的动力系统。许多这样的大系统是由偏微分方程的时变耦合系统的空间离散得到的。这种复杂动态系统的仿真对计算资源的要求非常高，当系统需要在许多不同的输入条件下进行查询时，这种高保真度的仿真可能会变得难以管理。本项目开发并应用了一类新的模型约简方法，用于动态系统的有效仿真和最优控制。本项目开发的模型简化方法使用较小的、计算效率高的模型来近似大型、复杂的时间相关过程模型，这些模型能够准确地表示原始过程在各种操作条件下的输出。因此，新的模型简化方法允许对系统进行仿真和控制，否则高保真计算模型将不实用。本项目开发的新的模型约简方法是数据驱动的。与现有方法不同，在许多情况下，新方法允许直接从（实验）数据生成高效的降阶计算模型，并且不需要访问高保真度计算模拟。总的来说，这种新方法提供了几个进步。首先，它生成一个对所有输入有效的原始非线性动力系统的降阶模型。其次，新方法是数据驱动的，避免了使用预测。这意味着不需要对原始系统的组件进行详细的访问，这使得新方法在无法获得系统仿真细节时特别有吸引力。该项目将开发用于创建此类数据驱动的降阶模型的数学算法，并提供其性能的理论分析。此外，本项目将新的模型约简方法集成到最优控制问题的求解中。对于这种集成，将确定关键的额外投入产出关系，并增加降阶模型的生成，以确保这些额外投入产出关系得到很好的近似。这种新方法将应用于重要生物现象和流体流动的模拟和控制。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。

Dynamical systems are a principal tool in the modeling, prediction, and control of physical phenomena ranging from heat dissipation in complex microelectronic devices, to vibration suppression in large wind turbines, to flow simulation. Optimal control of dynamical systems plays an important role in the many science and engineering applications where one wants to generate inputs to improve the performance of a system. However, ever-increasing need to include more detail at the modeling stage inevitably leads to larger-scale, more complex dynamical systems. Many of these large-scale systems are obtained from spatial discretizations of time-dependent coupled systems of partial differential equations. The simulation of such complex dynamical systems creates huge demands on computational resources and such high fidelity simulations may become unmanageable when the system needs to be queried at many different inputs. This project develops and applies a new class of model reduction methods for the efficient simulation and optimal control of dynamical systems. The model reduction approaches developed in this project approximate large, complex models of time-dependent processes using smaller, computationally efficient models that are nonetheless capable of representing accurately the outputs of the original process under a variety of operating conditions. Thus the new model reduction methods allow simulation and control of systems that would otherwise not be practical with high fidelity computational models.The new model reduction methods developed in this project are data-driven. Unlike existing methods, in many cases, the new methods allow the generation of efficient reduced order computational models directly from (experimental) data and do not require access to high fidelity computational simulations. Overall, this new approach offers several advancements. First, it generates a reduced order model of the original nonlinear dynamical system that is valid for all inputs. Second, the new approach is data-driven and avoids the use of projections. This means less detailed access to the components of the original system are needed, which makes the new method particularly attractive when details of system simulation is inaccessible. This project will develop mathematical algorithms for the creation of such data-driven reduced order models and provide theoretical analyses of their performance. In addition, this project integrates the new model reduction approach into the solution of optimal control problems. For this integration, crucial additional input-output relations will be identified and the reduced order model generation will be augmented to ensure that these additional input-output relations are well approximated. This new approach will be applied to simulation and control of important biological phenomena and of fluid flows.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.