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

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，动力系统是对各种物理现象进行建模、预测和控制的主要工具。动态系统的最优控制在许多科学和工程应用中扮演着重要的角色，在这些应用中，人们希望产生输入来改善系统的性能。然而，不断增长的在建模阶段包含更多细节的需求不可避免地导致了更大规模、更复杂的动态系统。这些大系统中的许多都是由依赖时间的偏微分方程组的空间离散得到的。这样的复杂动态系统的仿真产生了对计算资源的巨大需求，当需要在许多不同的输入下查询系统时，这样的高保真仿真可能变得无法管理。该项目开发和应用了一类新的模型降阶方法，用于动态系统的有效仿真和最优控制。本项目中开发的模型简化方法使用较小的、计算高效的模型来近似时间相关过程的大型、复杂模型，但这些模型仍能够在各种操作条件下准确地表示原始过程的输出。因此，新的模型降阶方法允许对高保真计算模型不实用的系统进行模拟和控制。本项目中开发的新模型降阶方法是数据驱动的。与现有方法不同，在许多情况下，新方法允许直接从(实验)数据生成有效的降阶计算模型，并且不需要访问高保真计算模拟。总体而言，这种新方法提供了几个进步。首先，它生成对所有输入都有效的原始非线性动力系统的降阶模型。其次，新的方法是数据驱动的，避免使用预测。这意味着需要对原始系统的组件进行不那么详细的访问，这使得当无法访问系统仿真的细节时，新方法特别有吸引力。该项目将为创建这种数据驱动的降阶模型开发数学算法，并对其性能进行理论分析。此外，该项目还将新的模型降阶方法融入到最优控制问题的求解中。对于这种集成，将识别关键的附加输入-输出关系，并将扩大降阶模型生成以确保这些附加的输入-输出关系被很好地逼近。这一新方法将应用于重要生物现象和流体流动的模拟和控制。这一奖项反映了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。

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.