Data-Driven Nonlinear Model Reduction with Applications to Fluid Flow Systems

这个动力学、控制和系统诊断（DCSD）项目将创建和研究系统模型约简算法，以捕捉非线性流体流动的显著特征。由于控制动力学方程的规模和复杂性，在制定控制流动行为的策略时，模型简化通常是必不可少的第一步。当非线性项占主导地位时，例如，在涉及高流速的应用中，现有的约简框架通常会失效。这些新技术将在各种应用中得到广泛应用，包括流动分离、提高升力和减少阻力。简化后的模型可用于飞机和运输油罐车的设计以及车辆排的控制。因此，这项研究的成果将促进国家的繁荣和国防。研究结果将纳入教育活动，以使来自代表性不足背景的学生受益。当非线性项主导动力学行为时，例如在高雷诺数时，许多现有的模型简化框架在流体流动应用中失效。该项目旨在通过开发模型简化算法来填补这一关键空白，该算法旨在捕捉非线性偏微分方程控制下流体流动中出现的基本非线性行为。考虑了两种主要方法。第一种方法研究了一个全局几何模型约简框架，该框架保留了固有的偏微分方程几何形状，并捕获了数据点对之间的测地线距离。第二种方法使用等稳坐标框架来描述控制周期或平稳解附近行为的最慢衰减非线性模式的潜在动力学。这两种策略都将使用快照数据来实现，以便它们可以很容易地应用于实验环境。该项目的成功完成将产生一个强大的通用框架，用于识别合适的降阶基，以分析偏微分方程驱动的流体流动系统中出现的基本非线性行为。原型问题描述的非线性对流越过障碍物，非定常气流在办公楼，和非定常流动的敏捷微型飞行器将被考虑。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 Dynamics, Control, and System Diagnostics (DCSD) project will create and investigate systematic model reduction algorithms to capture the salient features of nonlinear fluid flows. Due to the sheer scale and complexity of the governing dynamical equations, model reduction is often an imperative first step when formulating strategies to control flow behaviors. Existing reduction frameworks typically fail when nonlinear terms dominate, for instance, in applications involving high flow speeds. These new techniques will find use in a wide variety of applications, including those involving flow separation, lift enhancement, and drag reduction. The reduced models can be used for the design of aircraft and transport tankers and in the control of vehicular platoons. Consequently, results of this research will enhance national prosperity and defense. Research findings will be incorporated into educational activities to benefit students from underrepresented backgrounds.Many existing model reduction frameworks fail in fluid flow applications when nonlinear terms dominate the dynamical behavior, for instance, at high Reynolds numbers. This project aims to fill this critical gap by developing model reduction algorithms designed to capture fundamentally nonlinear behaviors that emerge in fluid flows governed by nonlinear partial differential equations. Two primary approaches are considered. The first approach investigates a global geometric model reduction framework that preserves the intrinsic partial differential equation geometry and captures geodesic distances between pairs of data points. The second uses an isostable coordinate framework that characterizes the underlying dynamics of the slowest decaying nonlinear modes that govern the behavior near either periodic or stationary solutions. Both strategies will be implemented using snapshot data so that they can be readily applied in experimental settings. Successful completion of this project will result in powerful, general frameworks for identifying suitable reduced order bases to analyze fundamentally nonlinear behaviors that emerge in partial differential equation driven fluid flow systems. Prototype problems describing nonlinear convection past obstacles, unsteady airflow in office buildings, and unsteady flows in agile micro-air vehicles will be considered.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.