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，这些参数捕捉流体通过剪切自身产生摩擦的能力，或量化流体内部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量来源，例如由于大规模破坏性火山爆发和人为温室气体效应而引起的大气扰动。这些参数的作用对于理解湍流、气候动力学和其他工程应用至关重要。它们的精确测定是通过收集数据经验得出的。通常，收集到的数据是不完整的，或者被测量过程或所用设备产生的噪声所损坏。该项目系统地解决了流体流动背景下参数的准确估计问题，通过数学分析和数据驱动方法的计算研究，旨在确定在几种适当理想化的实际情况下可证明保证参数恢复的条件。该项目还通过课程重新开发为利用数据革命建立基础设施，为本科生和研究生提供培训和指导机会，并通过同侪指导计划培养强大而多样化的数学文化。该项目将通过反馈控制范式对数据驱动的动态算法进行数学研究，以恢复流体动力学偏微分方程中出现的未知参数。在这种方法中，系统上的观测值作为外生项直接插入预测方程，以强制向观测值收敛。提出并系统分析了几种参数估计的实际情况，包括同时恢复多个参数和使用各种观测模式。支撑这一策略成功的一个关键特性是，在大量耗散系统中发现了一种已知的非线性机制，这种机制允许小规模信息逐渐被大规模信息所奴役。这一性质在参数估计中的应用是新颖的。因此，本研究将阐明该机制在流体流动以外的环境中参数估计的作用，并对该方法的鲁棒性和局限性进行全面探索。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 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.