Data-Driven Stochastic Model Reduction and Its Applications in Data Assimilation

随着数据的常规可用性，科学和工程中的许多应用要求及时和准确的预测，并不断更新。在数据同化中，这种预测是通过将数据与数学动态模型相结合来实现的。为了量化预测中的不确定性，需要对许多初始条件及时地重复求解动力学模型。这就排除了第一原理动力学模型的使用，因为计算成本高，求解耗时长，因此需要构建有效的统计动力学简化模型。本项目将基于统计推断、随机过程、动力系统和偏微分方程等方面的理论和工具，通过开发基于数据的随机约简模型来解决这一问题。本研究的基础是基于数据的计算建模和仿真，旨在开发新的数学和统计理论和工具，以解决复杂系统中基于数据的预测建模的挑战。该研究计划与教育目标相辅相成，通过跨学科研究为学生做好准备和培训。本项目的目标是开发和分析有效的算法，从复杂系统的数据中构建统计动态有效的随机约简模型，并利用这些约简模型通过数据同化获得及时的预测。研究者计划通过(i)耗散系统不变流形的参数化投影，以及（ii）通过一般未知系统的非参数推理方法近似未解决变量对已解决变量的影响，从数据中推断离散时间非马尔可夫简化模型。研究者还计划为这些非马尔可夫化简模型开发新的数据同化方法，并应用这些方法来解决从有噪声的部分数据中进行模型化简的巨大挑战。这项工作有望在气候建模、地球物理学、流体力学和其他数据信息计算建模问题中得到应用。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。

With data routinely available, many applications in science and engineering call for timely and accurate predictions that are constantly updated. In data assimilation, such predictions are made by combining data with mathematical dynamical models. To quantify the uncertainty in predictions, the dynamical models need to be repeatedly solved for many initial conditions in a timely manner. This rules out the use of first-principles dynamical models that are computationally expensive and time-consuming to solve, and prompts the need to construct effective statistical-dynamical reduced models. This project will address this issue by developing data-informed stochastic reduced models based on theories and tools in statistical inference, stochastic processes, dynamical systems, and partial differential equations. The research lies at the foundation of data-informed computational modeling and simulation, and aims at developing new mathematical and statistical theories and tools to address the challenges in data-informed predictive modeling of complex systems. The research plan is complemented by educational objectives to prepare and train students through interdisciplinary research. The goal of this project is to develop and analyze efficient algorithms to construct statistically-dynamically effective stochastic reduced models from data for complex systems, and to obtain timely predictions through data assimilation using these reduced models. The investigator plans to infer discrete-time non-Markovian reduced models from data by (i) parametrizing projections of invariant manifolds of dissipative systems, and (ii) approximating the effects of the unresolved variables on the resolved variables by nonparametric inference methods for general unknown systems. The investigator also plans to develop novel data assimilation methods for these non-Markovian reduced models and apply these methods to address the grand challenge of model reduction from noisy partial data. The work is expected to have applications in climate modeling, geophysics, fluid mechanics, and other data-informed computational modeling problems.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.