Manifold learning and deep autoencoders for nonlinear embedding of unsteady fluid flows

Computational fluid dynamics (CFD) is known for producing high-dimensional data in space and time. Modern data-driven modeling approaches present a myriad of techniques to extract physical information from these datasets and identify an optimal set of coordinates for representing them in a low-dimensional embedding. This is a crucial first step toward reduced order modeling, usually done via proper orthogonal decomposition (POD), which gives the best linear approximation. However, fluid flows are often highly complex with nonlinear structures. Several unsupervised machine learning algorithms have been developed in other branches of science for nonlinear dimensionality reduction (NDR), but have not yet been extensively used for fluid flow data. We investigate four manifold learning and two deep learning based NDR methods and compare them to POD. These are tested on two canonical fluid flow problems and biomedical flows in diseased arteries. We compare the performance of these methods and discuss the associated challenges. The temporal vs. spatial arrangement of input data and its influence on NDR mode extraction is investigated, and the obtained spatial modes are compared. Finite time Lyapunov exponents (FTLE) are calculated to facilitate flow physics interpretation.