LEAPS-MPS: Dynamical Parameter Estimation for Hydrodynamic Equations

该奖项全部或部分由2021年美国救援计划法案（公法117-2）资助。物理现象建模的一个基本特征是存在对各种相关属性进行编码的参数。在流体流动的研究中，出现了一些参数，这些参数捕获了流体通过剪切自身产生摩擦的能力，或者量化了流体内的温差如何驱动其运动，例如当太阳加热海洋和火焰加热一壶水时，或者更一般地表示外部能源，例如大规模破坏性火山爆发和人为温室气体效应造成的大气扰动。这些参数的作用是至关重要的湍流，气候动力学和其他工程应用的理解。它们的精确测定是通过收集数据凭经验获得的。通常，收集的数据不完整或被测量程序或所使用设备可能产生的噪音破坏。该项目系统地解决了流体流动中参数的准确估计问题，通过对数据驱动方法进行数学分析和计算研究，以确定在几种适当理想化的实际情况下可以证明保证恢复参数的条件。该项目还通过课程重建建立了利用数据革命的基础设施，为本科生和研究生提供培训和指导机会，并通过同行指导计划培养了强大而多样化的数学文化。该项目将对数据驱动的动态算法进行数学研究，以便通过反馈控制范例恢复流体力学偏微分方程中出现的未知参数。在这种方法中，对系统的观察直接插入到预测方程作为一个外源项，强制收敛到观察。几个实际情况下的参数估计的设想和系统的分析，其中包括多个参数的同时恢复和使用各种模式的观测。支撑这一战略的成功的一个关键属性是存在一个已知的非线性机制，发现在一个大类的耗散系统，允许小规模的信息渐近奴役大规模的信息。这一特性在参数估计中的应用是新颖的。因此，这项研究将阐明这种机制的作用，参数估计的背景下超越流体流动，并提供了一个全面的探索的鲁棒性和局限性，这种approach.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).A fundamental feature in the modeling of physical phenomena is the presence of parameters that encode various relevant properties. In the study of fluid flows, parameters arise that capture the ability of the fluid to generate friction by shearing against itself or quantify how temperature differences within the fluid can drive its motion, such as when the sun heats the ocean and a flame heats a pot of water, or more generally represent external sources of energy, such as disturbances in the atmosphere due to large-scale disruptive volcanic eruptions and anthropogenic greenhouse gas effects. The role of these parameters is crucial in the understanding of turbulence, climate dynamics, and other engineering applications. Their precise determination is obtained empirically through the collection of data. Often the data collected is incomplete or corrupted by noise that may arise from the measurement procedure, or the devices used. This project systematically addresses the issue of the accurate estimation of parameters in the context of fluid flows, through the mathematical analysis and computational study of a data-driven approach geared to identify conditions under which recovery of the parameters can be provably guaranteed in several suitably idealized practical situations. The project also builds infrastructure for harnessing the data revolution through course redevelopment, provides training and mentoring opportunities for undergraduate and graduate students, and fosters a robust and diverse mathematical culture through a peer mentorship program. The project will carry out a mathematical study of a data-driven dynamical algorithm for recovering unknown parameters arising in partial differential equations of hydrodynamics via a paradigm of feedback control. In this approach, observations on the system are directly inserted into the prognostic equation as an exogenous term that enforces convergence towards the observations. Several practical scenarios of parameter estimation are conceived and systematically analyzed, which include the simultaneous recovery of multiple parameters and the use of various modes of observations. A key property underpinning the success of this strategy is the existence of a known nonlinear mechanism found in a large class of dissipative systems that allows small scale information to be asymptotically enslaved to large scale information. The application of this property to parameter estimation is novel. As a result, this study will shed light on the role of this mechanism for parameter estimation in contexts beyond fluid flow and provide a comprehensive exploration of the robustness and limitations of this approach.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.