This article introduces an advanced Koopman mode decomposition (KMD) technique -- coined Featurized Koopman Mode Decomposition (FKMD) -- that uses time embedding and Mahalanobis scaling to enhance analysis and prediction of high dimensional dynamical systems. The time embedding expands the observation space to better capture underlying manifold structure, while the Mahalanobis scaling, applied to kernel or random Fourier features, adjusts observations based on the system's dynamics. This aids in featurizing KMD in cases where good features are not a priori known. We find that the Mahalanobis scaling from FKMD can be used for effective dimensionality reduction of alanine dipeptide data. We also show that FKMD improves predictions for a high-dimensional Lorenz attractor and a cell signaling problem from cancer research.

翻译：本文提出一种先进的库普曼模式分解（KMD）技术——称为特征化库普曼模式分解（FKMD），该技术利用时间嵌入和马氏距离缩放来增强高维动力系统的分析与预测。时间嵌入扩展了观测空间以更好地捕捉潜在流形结构，而应用于核函数或随机傅里叶特征的马氏距离缩放则基于系统动力学调整观测数据。这有助于在事先未知良好特征的情况下实现KMD的特征化。我们发现FKMD中的马氏距离缩放可有效降低丙氨酸二肽数据的维度。同时，FKMD还改进了高维洛伦兹吸引子及癌症研究中的细胞信号传导问题的预测效果。