CDS&E-MSS: Predictive Modeling and Data-Driven Closure of Chaotic and Noisy Dynamics in Discrete Time

从人脑到电网，科学家和工程师越来越依赖大规模计算机模型来理解、预测甚至控制复杂的动态系统。然而，这些模型往往反映了它们所代表的系统的复杂性，并且需要超级计算机或大型计算集群来运行。对于现代科学计算的许多任务来说，这可能是一个障碍。例如，在基于物理测量估计模型参数值时(使用模型进行预测的关键步骤)，需要多次运行模型。即使计算能力不断增强，这也可能很耗时。这个项目研究计算和数学技术，以构建更简单的模型，同时捕捉更复杂模型的关键特征，以便可以更有效地完成参数估计等任务。这项研究产生的算法有望适用于广泛的科学和工程问题。该研究项目也有望成为青年科学家进行跨学科研究的一种很好的培训工具。更详细地，该项目涉及高维混沌/噪声动力系统的离散时间降阶方法。其主要目标是：(1)建立一个离散时间模型简化的通用数学框架，侧重于实现短期和中期预测以及复制选定的长期统计数据；(2)调整通用方法，以处理实际中出现的特定类型的动力系统，例如，混沌、噪声等；(3)将该方法应用于特定生物学和物理应用中的模型简化和数据驱动的建模问题。由于当代科学和工程中的许多动力学系统都是由跨越空间和时间尺度的大量自由度相互作用组成的，所以详细的第一性原理模型可能不能为不确定性量化、参数/状态估计和优化等任务提供实际基础。此外，对于许多这样的系统，我们对相关事实和原则的了解是不完整的。因此，简化的模型在不分解所有自由度的情况下捕捉到了本质的动力学，因此具有实用价值。他们对自己的权利也非常感兴趣，因为好的简化模型忽略了不相关的细节，往往包含对手头现象的有用见解。该项目借鉴了应用概率、动力系统和统计物理的不同观点，并为培训掌握这些工具的数学科学家提供了充足的机会。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

From the human brain to the electric power grid, scientists and engineers increasingly depend on large-scale computer models to understand, predict, and even control complex dynamical systems. However, these models often reflect the complexity of the systems they represent, and require supercomputers or large computing clusters to run. For many tasks of modern scientific computing, this can represent an impediment. For example, when estimating the values of model parameters based on physical measurements (a critical step in using a model to make predictions), it is necessary to run a model many times. Even with ever-increasing computational power, this may be time-consuming. This project studies computational and mathematical techniques for constructing simpler models that nevertheless capture key features of more complex models, so that tasks like parameter estimation can be done more efficiently. The algorithms resulting from this research are expected to be applicable to a wide range of scientific and engineering problems. This research project is also expected to be a good training vehicle for young scientists in interdisciplinary research.In more detail, this project concerns a discrete-time approach to model reduction for high-dimensional chaotic / noisy dynamical systems. The primary aims are (i) to develop a general mathematical framework for discrete-time model reduction, focusing on enabling both short- to medium-range forecasting as well as reproducing selected long-time statistics; (ii) adapting the general methods to handle specific types of dynamical systems that arise in practice, e.g., chaotic, noisy, etc.; (iii) apply the methodology to model reduction and data-driven modeling problems in specific biological and physical applications. As many dynamical systems of interest in contemporary science and engineering consist of a large number of degrees of freedom interacting across spatial and temporal scales, detailed first-principles models may not provide a practical basis for tasks like uncertainty quantification, parameter / state estimation, and optimization. Moreover, for many such systems, our knowledge of pertinent facts and principles are incomplete. Reduced models that capture the essential dynamics without resolving all degrees of freedom are thus of practical utility. They are also of great interest in their own right, as good reduced models leave out irrelevant details and often contain useful insights into the phenomenon at hand. This project draws on diverse ideas from applied probability, dynamical systems, and statistical physics, and provide ample opportunities for training mathematical scientists for the mastery of these tools.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.