Collaborative Research: Data Assimilation for Turbulent Flows: Dynamic Model Learning and Solution Capturing

这个世界充满了复杂的、多尺度的现象，由于其潜在的混沌性质，预测这些现象可能具有挑战性。 例如，快速准确地预测天气现象（包括陆地和太阳）、海洋动力学和地下水流对经济增长和稳定至关重要。 这些预测通常包含数学模型的计算机模拟;然而，为了做出准确的预测，这些模型需要正确地“初始化”;也就是说，它们需要非常精确地知道系统的当前状态，并且需要基于对所讨论系统的实际观察来调整模型。 例如，为了准确地预测天气，模型通常需要在几英寸的尺度上了解天气的当前状态，但天气观测站通常相隔几英里。 由于天气是一种高度混沌的现象，观测中的小误差和/或实际观测中的稀疏性可能导致预测中的重大错误。 为了解决这一问题，在过去几十年中开发了一系列称为“数据同化”的技术。 数据同化将观测数据纳入感兴趣系统的数学模型，以使预测达到正确的状态。 然而，标准的数据同化技术，被称为卡尔曼滤波器和四维变分（4D-VAR）的方法，是非常昂贵的计算，和重大的挑战仍然存在时，使其适应复杂的系统。 最近，一种新的数据同化算法，被称为Azouani-Olson-Titi（AOT）算法已经成为一种快速，鲁棒，高精度的技术，它很容易适应各种各样的模型，这是计算成本低廉，添加到一个已经存在的计算模型。 该项目不仅将扩展和改进AOT算法，还将使用PI和合著者发明的新思想和技术来调整AOT框架，以了解更多关于底层数学模型本身的信息，进一步提高预测能力。 该项目促进指导本科生和研究生，跨学科研究，并与国家实验室的互动。 该项目的影响将是深远的，将铺平道路的新技术，这将大大加快数据同化模拟高度复杂的流体流动，引入新的技术参数学习和模型重建，并提供一个计算方法来调查基本的数学问题。计算技术和数学工具的发展将是有用的科学家和工程师在其他领域以及。这个项目建立在以前的工作PI的AOT算法，这表明，这种算法可以适应学习（未知）参数的系统，甚至模型本身的形式，同时恢复的“真实”状态的系统。初步工作的扩展将进行，并严格的理由收敛的算法将完成物理有趣的系统，包括嘈杂的数据和稀疏的时间观测。此外，AOT本身的几个扩展将进行数值测试和严格的研究：间歇性观测的轻推，以及基于移动观察者的轻推。AOT还将用于地球海洋的多物理场大尺度模型、土壤湿度的理查兹方程以及使用实时收集的数据进行的简化流体实验。该项目将优化观测要求，以提高精度，大幅降低成本。这里描述的方法有可能降低生产和计算成本的实验，使他们更有用的研究人员在真实的世界的问题。此外，还将开发新的证明方法，以证明非线性AOT算法、基于移动观测器的AOT、基于AOT的模型恢复、基于温度的AOT以及将AOT扩展到地球物理环境等情况下的收敛性。该奖项反映了NSF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。

The world is full of complex, multi-scale phenomena that can be challenging to predict due to their underlying chaotic nature. For example, fast and accurate predictions of weather phenomena (both terrestrial and solar), ocean dynamics, and groundwater flow are vital to economic growth and stability. These predictions typically incorporate computer simulations of mathematical models; however, to make accurate predictions, these models need to be properly "initialized"; that is, they need to know the current state of the system very precisely and the models need to be adjusted based on actual observations of the system in question. For example, in order to accurately predict the weather, models often require the current state of the weather to be known on the scale of a few inches, but weather observation stations are often spaced several miles apart. Since weather is a highly chaotic phenomenon, small errors in the observations and/or sparsity in the actual observations can lead to significant errors in the predictions. To address this issue in the past several decades a collection of techniques known as "data assimilation" have been developed. Data assimilation incorporates observational data into the mathematical model of the system of interest in order to drive the prediction to the correct state. However, the standard data assimilation techniques, known as the Kalman filter and four-dimensional variational (4D-VAR) approaches, are very computationally costly, and major challenges still exist when adapting them to complex systems. Recently, a new algorithm for data assimilation, known as the Azouani-Olson-Titi (AOT) algorithm has emerged as a fast, robust, highly accurate technique which is easy to adapt to a wide variety of models, and which is computationally inexpensive to add to an already existing computational model. This project will not only extend and improve the AOT algorithm, but it will also use new ideas and technologies invented by the PIs and coauthors to adapt the AOT framework to learn more about the underlying mathematical model itself, further improving predictive capabilities. This project fosters mentoring undergraduate and graduate students, interdisciplinary research, and interaction with national labs. The impacts of this project will be far-reaching and will pave the way for new techniques which will greatly speed up data assimilation in simulations of highly complicated fluid flows, introduce novel techniques for parameter learning and model reconstruction, and provide a computational approach to investigating fundamental mathematical problems. The computational technologies and mathematical tools developed will be useful to scientists and engineers in other fields as well.This project builds on previous work of the PIs on the AOT algorithm, which showed that this algorithm can be adapted to learn the (unknown) parameters of the system, and even the form of the model itself, while simultaneously recovering the "true" state of the system. Extensions of the preliminary work will be carried out, and rigorous justification for convergence of the algorithm will be completed for physically interesting systems, including noisy data and sparse-in-time observations. In addition, several extensions of AOT itself will be numerically tested and rigorously investigated: nudging for intermittent observations, as well as nudging based on moving observers. AOT will also be implemented and tested for a multi-physics large-scale model of the Earth's oceans, for the Richards equation for soil moisture, and for a simplified fluids experiment using real-time collected data. This project will optimize observer requirements for better accuracy, significantly lowering costs. The methods described here have the potential to reduce production and computational cost for experiments, making them more useful to researchers working on real world problems. Moreover, novel proof methods will be developed to prove convergence in the cases of nonlinear AOT algorithms, AOT-based on moving observers, AOT-based model recovery, temperature-based AOT, and extensions of AOT to geophysical settings.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.