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

世界上充满了复杂的、多尺度的现象，由于其潜在的混乱性质，这些现象可能难以预测。例如，对天气现象(陆地和太阳)、海洋动力和地下水流动的快速准确预测对经济增长和稳定至关重要。这些预测通常包括对数学模型的计算机模拟；然而，为了做出准确的预测，这些模型需要适当地“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，并且需要根据对所述系统的实际观察来调整模型。例如，为了准确预测天气，模型往往要求在几英寸的尺度上知道天气的当前状态，但天气观测站往往相距几英里。由于天气是一种高度混乱的现象，观测中的微小误差和/或实际观测中的稀疏性可能会导致预测中的重大误差。为了解决这个问题，在过去的几十年里，已经开发了一套称为“数据同化”的技术。数据同化将观测数据合并到感兴趣系统的数学模型中，以便将预测推向正确的状态。然而，标准的数据同化技术，即卡尔曼滤波和四维变分(4D-VAR)方法，计算成本非常高，在使它们适应复杂系统时仍然存在重大挑战。最近，一种被称为Azouani-Olson-Titi(AOT)算法的新的数据同化算法已经出现，它是一种快速、健壮、高精度的技术，易于适应各种不同的模式，并且添加到现有的计算模式中计算成本较低。该项目不仅将扩展和改进AOT算法，还将使用PIS和合著者发明的新思想和技术来调整AOT框架，以更多地了解基础数学模型本身，进一步提高预测能力。该项目促进了对本科生和研究生的指导、跨学科研究以及与国家实验室的互动。该项目的影响将是深远的，并将为新技术铺平道路，这些技术将大大加快高度复杂流体流动模拟中的数据同化，引入新的参数学习和模型重建技术，并为研究基本数学问题提供计算方法。开发的计算技术和数学工具也将对其他领域的科学家和工程师有用。该项目建立在PIS先前关于AOT算法的工作基础上，该算法可以适应学习系统的(未知)参数，甚至模型本身的形式，同时恢复系统的“真实”状态。将进行前期工作的扩展，并将针对物理上感兴趣的系统，包括噪声数据和稀疏时间观测，完成算法收敛的严格证明。此外，将对AOT本身的几个扩展进行数值测试和严格调查：间歇观测的轻推，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大规模模型、土壤湿度的Richards方程以及使用实时收集的数据进行简化的流体实验。该项目将优化观察者的要求，以提高准确性，显著降低成本。这里描述的方法有可能减少实验的生产和计算成本，使它们对研究现实世界问题的研究人员更有用。此外，将开发新的证明方法来证明在非线性AOT算法、基于移动观察者的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的情况下的收敛。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.