Constructing sparse, effective reduced-order models (ROMs) for high-dimensional dynamical data is an active area of research in applied sciences. In this work, we study an efficient approach to identifying such sparse ROMs using an information-theoretic indicator called causation entropy. Given a feature library of possible building block terms for the sought ROMs, the causation entropy ranks the importance of each term to the dynamics conveyed by the training data before a parameter estimation procedure is performed. It thus allows for an efficient construction of a hierarchy of ROMs with varying degrees of sparsity to effectively handle different tasks. This article examines the ability of the causation entropy to identify skillful sparse ROMs when a relatively high-dimensional ROM is required to emulate the dynamics conveyed by the training dataset. We demonstrate that a Gaussian approximation of the causation entropy still performs exceptionally well even in presence of highly non-Gaussian statistics. Such approximations provide an efficient way to access the otherwise hard to compute causation entropies when the selected feature library contains a large number of candidate functions. Besides recovering long-term statistics, we also demonstrate good performance of the obtained ROMs in recovering unobserved dynamics via data assimilation with partial observations, a test that has not been done before for causation-based ROMs of partial differential equations. The paradigmatic Kuramoto-Sivashinsky equation placed in a chaotic regime with highly skewed, multimodal statistics is utilized for these purposes.

翻译：为高维动态数据构建稀疏且有效的降阶模型是应用科学领域的一个活跃研究方向。本文研究了一种利用因果熵这一信息论指标来识别此类稀疏降阶模型的高效方法。在给定目标降阶模型可能构建项的特征库后，因果熵能在参数估计程序执行前，根据训练数据所蕴含的动力学特性对各项的重要性进行排序。因此，该方法能够高效构建具有不同稀疏度的降阶模型层次结构，以有效处理不同任务。本文考察了当需要相对高维的降阶模型来模拟训练数据集所呈现的动力学时，因果熵识别高性能稀疏降阶模型的能力。我们证明，即使存在高度非高斯统计特性，因果熵的高斯近似仍然表现优异。当所选特征库包含大量候选函数时，此类近似为计算原本难以处理的因果熵提供了有效途径。除了恢复长期统计特性外，我们还通过部分观测数据同化证明了所得降阶模型在恢复未观测动力学方面的良好性能，这是此前基于因果关系的偏微分方程降阶模型尚未进行过的测试。为此，我们采用处于混沌状态且具有高度偏态多峰统计特性的典范Kuramoto-Sivashinsky方程进行验证。