Convolutional autoencoders, clustering, and POD for low-dimensional parametrization of flow equations

Simulations of large-scale dynamical systems require expensive computations and large amounts of storage. Low-dimensional representations of high-dimensional states such as in reduced order models deriving from, say, Proper Orthogonal Decomposition (POD) trade in a reduced model complexity against accuracy and can be a solution to lessen the computational burdens. However, for really low-dimensional parametrizations of the states as they may be needed for example for controller design, linear methods like the POD come to their natural limits so that nonlinear approaches will be the methods of choice. In this work, we propose a convolutional autoencoder (CAE) consisting of a nonlinear encoder and an affine linear decoder and consider a deep clustering model where a CAE is integrated with k-means clustering for improved encoding performance. The proposed set of methods is compared to the standard POD approach in three scenarios: single- and double-cylinder wakes modeled by incompressible Navier-Stokes equations and flow setup described by viscous Burgers' equations.