Basis Identification for Nonlinear Dynamical Systems Using Sparse Coding

Abstract

Basis identification is a critical step in the construction of accurate reduced order models using Galerkin projection. This is particularly challenging in unsteady nonlinear flow fields due to the presence of multi-scale phenomena that cannot be ignored and are not well captured using the ubiquitous Proper Orthogonal Decomposition. This study focuses on this issue by exploring an approach known as sparse coding for the basis identification problem. Compared to Proper Orthogonal Decomposition, which seeks to truncate the basis spanning an observed data set into a small set of dominant modes, sparse coding is used to select a compact basis that best spans the entire data set. Thus, the resulting bases are inherently multi-scale, enabling improved reduced order modeling of unsteady flow fields. The approach is demonstrated for a canonical problem of an incompressible flow inside a 2-D lid-driven cavity. Results indicate that Galerkin reduction of the governing equations using sparse modes yields significantly improved fluid predictions.