Machine learning of fluid flows using the proper orthogonal decomposition (POD) and the dynamic mode decomposition (DMD) methods 1. Introduction 1.1 Background and Motivation Fluid dynamics has long been challenging due to its inherent high-dimensionality and nonlinear behaviour. Traditional numerical approaches, while effective, often become computationally prohibitive when applied to complex flows, such as turbulent flows past a cylinder. In recent years, data-driven methods have emerged as promising alternatives by extracting the essential dynamics directly from high-fidelity simulation or experimental data. Two techniques, particularly Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD), have gained prominence for their ability to identify coherent structures and predict temporal evolution in fluid flows. POD, closely related to the singular value decomposition (SVD), decomposes complex flow fields into a set of orthogonal modes ordered by energy conte...