EAGER-DynamicData: Collaborative Research: Data-driven morphing of parsimonious models for the description of transient dynamics in complex systems

在工程和科学中，预测和量化复杂系统的行为对于设计、优化和安全等许多领域都是一个至关重要的课题。更关键的是对这些系统极端反应的预测。可能导致灾难性事件的罕见反应表现在广泛的系统中，例如地球物理现象、电力和通信网络以及大脑活动中的癫痫事件，仅举几例。在所有这些情况下，准确的预测都受到这样一个事实的阻碍，即自然界中系统的确切动力学通常很难理解。这是由于在不同的时间和空间尺度上运作的大量本质上相互耦合的机制造成的。尽管在所有这些不同的级别上高精度地预测系统并不总是必不可少的，但重要的是了解未解决的机制对我们想要预测的自由度的影响并对其进行建模。这需要对这些机制的描述性规律以及它们与兴趣自由度的耦合有可靠的了解，而这显然并不总是(很好)做到。这种对动力学的不完全建模导致了不可避免的模型误差，这对于可靠的预测是必不可少的。实现这一点的一种显而易见的方法是利用可用的和动态传入的数据。这项工作的目标是开发方法和算法来扩展简约模型的数据驱动变形(即数据驱动适应)的能力。这些将能够充分地捕捉系统的瞬时最重要的动态，并利用它们来廉价地执行信息性预测和不确定性量化。这一发展对于建模和预测能力因对潜在物理机制的了解不足而受到限制的许多领域将是至关重要的。该建议的目的是在数据流驱动的环境中将机器学习与模型简化联系起来，以便为复杂随机系统的概率预测制定全新的方法。特别令人感兴趣的是对极端反应的量化和预测，完全依靠现有数据的利用，并尽可能少地使用方程式(或高保真解算器)(如果有的话)。这一努力是由与预测复杂动力系统中的这些特征相关的严重障碍的存在推动的：描述律中不可忽略的模型误差(如果这些可用)、实时计算的高昂成本、稀疏数据或具有不可忽略误差的数据、以及瞬时动力学。在气候动力学、非线性波浪和高维网络等背景下非常需要理解和预测极端反应的时候，这些困难是显而易见的。其目的是通过构建新的随机预测方法来解决这些挑战，所述新的随机预测方法将通过实施(和适当扩展)机器学习/数据挖掘技术以及与根据动态动态地自适应降阶子空间的随机降阶和不确定性量化方法的组合来扩展现有的数据驱动建模和预测的最新水平。这些工作将由一个概念验证应用程序指导，该应用程序涉及预测非线性波中的极端局部事件。由数据驱动的自适应降阶模型将是推断关键动力学特性的关键元素，否则这些特性将“埋藏”在复杂的响应中。通过将机器学习技术与自适应降阶模型联系起来，我们的研究将催生数值/数学分析的新领域，并将扩展更传统的数学辅助建模的范围，超越其目前的一些限制。

Predicting and quantifying the behavior of complex systems in engineering and science is a topic of critical importance for many areas such as design, optimization, and safety. Even more critical is the forecast of extreme responses of these systems. Rare responses that can lead to catastrophic events manifest themselves in a wide range of systems such as geophysical phenomena, power and communication networks, and epileptic incidents in brain activity, just to mention a few. In all of these cases, accurate predictions are hampered by the fact that the exact dynamics of the system in nature is often poorly understood. This poor understanding is due to the large number of essentially coupled mechanisms that operate at different temporal and spatial scales. Although it is not always essential to predict the system with high accuracy over all these different levels, it is important to understand and model the effect of the unresolved mechanisms to the degrees of freedom we want to predict. This requires a reliable knowledge of the descriptive laws for these mechanisms as well as their coupling to the degrees-of-freedom of interest and this is clearly not always (well) done. This incomplete modeling of the dynamics leads to inevitable model error that is essential to be taken into account for reliable predictions. An obvious way that this can be done is by the utilization of available and dynamically incoming data. The goal of this work is the development of methods and algorithms to extend the capability for data-driven morphing (that is, data-driven adaptation) of parsimonious models. These will be able to adequately capture the instantaneously most significant dynamics of the system and utilize them to inexpensively perform informative prediction and uncertainty quantification. Such a development will be of critical importance for many fields where the modeling and predictive capacity is limited by the inadequate understanding of the underlying physical mechanisms.The aim of this proposal is to link machine learning with model reduction in a data-stream-driven environment, in order to formulate fundamentally novel methods for the probabilistic forecast of complex stochastic systems. Of particular interest is the quantification and prediction of extreme responses, by relying exclusively on the utilization of available data and with the minimum use of equations (or high fidelity solvers), if these are available. The effort is driven by the presence of serious obstacles associated with the prediction of such features in complex dynamical systems: non-negligible model error in the descriptive laws (if these are available), prohibitive cost for real time computations, sparse data or data with non-negligible error, and transient dynamics. These difficulties are manifest at a time when there is a great need for understanding and prediction of extreme responses in contexts such as climate dynamics, nonlinear waves, and networks of high dimensionality. The aim is to address several of theses challenges by constructing new stochastic prediction methods that will extend the existing state-of-the-art for data-driven modeling and prediction through the implementation (and appropriate extension) of machine learning / data mining techniques and the combination with stochastic order reduction and uncertainty quantification methods that dynamically adapt the reduced order subspace according to the dynamics. These efforts will be guided by a proof-of-concept application involving prediction of extreme, localized events in nonlinear waves. Adaptive reduced order models driven by data will be a key element for the inference of critical dynamical properties, which are otherwise "buried" in the complex responses. By linking machine learning techniques to adaptive reduced order models our research will catalyze new domains of numerical/mathematical analysis and it will extend the reach of more conventional mathematics-assisted modeling beyond some of its current limits.