Enhancing the sparsity of data-driven reduced-order models (ROMs) has gained increasing attention in recent years. In this work, we analyze an efficient approach to identifying skillful ROMs with a sparse structure using an information-theoretic indicator called causation entropy. The causation entropy quantifies in a statistical way the additional contribution of each term to the underlying dynamics beyond the information already captured by all the other terms in the ansatz. By doing so, the causation entropy assesses the importance of each term to the dynamics before a parameter estimation procedure is performed. Thus, the approach can be utilized to eliminate terms with little dynamic impact, leading to a parsimonious structure that retains the essential physics. To circumvent the difficulty of estimating high-dimensional probability density functions (PDFs) involved in the causation entropy computation, we leverage Gaussian approximations for such PDFs, which are demonstrated to be sufficient even in the presence of highly non-Gaussian dynamics. The effectiveness of the approach is illustrated by the Kuramoto-Sivashinsky equation by building sparse causation-based ROMs for various purposes, such as recovering long-term statistics and inferring unobserved dynamics via data assimilation with partial observations.

翻译：近年来，增强数据驱动降阶模型（ROMs）的稀疏性日益受到关注。本研究分析了一种利用信息论指标——因果熵——来识别具有稀疏结构的有效降阶模型的方法。因果熵以统计方式量化了基函数中每个项对底层动力学的额外贡献，该贡献超越了基函数中所有其他项已捕获的信息。通过这种方式，因果熵在参数估计过程执行之前评估了每个项对动力学的重要性。因此，该方法可用于消除对动力学影响甚微的项，从而得到一个保留基本物理规律的简约结构。为了规避因果熵计算中涉及的高维概率密度函数（PDFs）估计的困难，我们对此类PDFs采用高斯近似，该近似被证明即使在存在高度非高斯动力学的情况下也是充分的。通过Kuramoto-Sivashinsky方程，我们构建了基于因果关系的稀疏降阶模型，用于恢复长期统计特性以及通过部分观测数据同化推断未观测动力学等多种目的，从而验证了该方法的有效性。