Deep Learning-aided Spectral Analysis of Fluid Flows

A linear time-invariant (LTI) representation of Markovian dynamics through the Koopman operator is often leveraged for characterizing the spectral behavior of the dynamical systems. The key to the success of the Koopman-based representation and the relevance of the learned spectral characteristics to the underlying system depends on the appropriateness of the map that transforms the input state to the 'observable'. In this work, we leverage deep learning algorithms to deduce transformation maps from training data and ultimately the LTI Koopman operator. In particularly, we present (i) a deep Koopman network (DKN) that learns the observable and the transition operator simultaneously in a single network and (ii) a deep autoencoder network (DAN) that accomplishes the same in two steps. We compare the efficacy of these architectures to the class of Dynamic Mode Decomposition (DMD)-based Koopman approximation techniques. While deep learning-based methods are shown to be good at capturing temporally evolving dynamics for systems that can be represented by limited data. In this talk, we will focus on the accuracy of the spectral information, like physically important eigenmodes and eigenvalues found using these methods from canonical PDE systems and fluids flows.