Comparative Analysis of Manifold Learning Techniques for Controlled flows*

Turbulent flows, despite their high dimensionality, exhibit recurrent patterns known as coherent structures, suggesting the possibility of representing key dynamics on a low-dimensional manifold. Manifold learning aims to identify such low-dimensional surfaces. Farzamnik et al. (2023, J Fluid Mech, 955:A34) demonstrated the effectiveness of low-dimensional manifold learning in describing shear flows. However, control inputs alter flow dynamics, complicating manifold identification. This work compares the performance of key data-driven manifold learning techniques for controlled flows, including standard and kernel Principal Component Analysis (kPCA), Multidimensional Scaling (MDS), Isometric Feature Mapping (ISOMAP), and Locally Linear Embedding (LLE). The test case is the fluidic pinball, a configuration of three cylinders in a uniform flow with independently controlled rotation. Despite its relatively simple dynamics, it offers a wide range of control possibilities, allowing an extensive span of different flow configurations. The results reveal that nonlinear methods capture meaningful coordinates in controlled flows, which can be used for low-order modeling.