Approximate Signal Identification in Pattern Recognition Using Dynamic Mode Decomposition (DMD) and Koopman theory

Abstract

In this work we apply a signal identification system that uses the dynamic mode decomposition algorithm (DMD), it is based on proper orthogonal decomposition (POD), which uses computationally efficient singular value decomposition (SVD) so that it scales good for providing dimensionality reduction in high dimension systems. DMD is a technique that allows the extraction of dynamically relevant flow characteristics from experimental or numerical data, for which a comparison is made with the principal component analysis (PCA) technique and it is shown that the results obtained by DMD are very good. We also consider the application of Koopman’s theory to Schrödinger’s nonlinear partial differential equation where we show that the observables chosen to construct the Koopman operator are fundamental to allow a good approximation to nonlinear dynamics. If such observables can be found, then the dynamic mode decomposition algorithm can be applied to compute a finite-dimensional approximation of the Koopman operator, including its eigenfunctions, eigenvalues, and Koopman modes.