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于大型计算机模型来理解、预测甚至控制复杂的动力系统。然而，这些模型通常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，当基于物理测量估计模型参数的值时（使用模型进行预测的关键步骤），需要多次运行模型。即使计算能力不断提高，这也可能是耗时的。该项目研究计算和数学技术，用于构建更简单的模型，但捕获更复杂模型的关键特征，以便更有效地完成参数估计等任务。从这项研究中产生的算法预计将适用于广泛的科学和工程问题。该研究项目也有望成为一个很好的培训工具，为年轻的科学家在跨学科的研究。更详细地说，该项目涉及到离散时间的方法来减少高维混沌/噪声动力系统的模型。主要目标是（i）为离散时间模型简化开发一个通用数学框架，重点是实现中短期预测以及再现选定的长期统计数据;（ii）调整通用方法以处理实践中出现的特定类型的动力系统，例如，混乱、嘈杂等; (iii)将该方法应用于特定生物和物理应用中的模型简化和数据驱动建模问题。由于当代科学和工程中许多感兴趣的动力系统由大量的跨空间和时间尺度相互作用的自由度组成，详细的第一性原理模型可能无法为不确定性量化，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理的知识是不完整的。简化模型，捕捉基本的动态，而不解决所有的自由度，因此具有实用价值。它们本身也很有趣，因为好的简化模型会忽略不相关的细节，并且通常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学的各种思想，并为培训数学科学家掌握这些工具提供了充足的机会。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值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.