Approximation of translation invariant Koopman operators for coupled non-linear systems

Many physical systems exhibit translational invariance, meaning that the underlying physical laws are independent of the position in space. Data driven approximations of the infinite dimensional but linear Koopman operator of non-linear dynamical systems need to be physically informed in order to respect such physical symmetries. In the current work, we introduce a translation invariant extended dynamic mode decomposition (tieDMD) for coupled non-linear systems on periodic domains. This is achieved by exploiting a block-diagonal structure of the Koopman operator in Fourier space. Variants of tieDMD are applied to data obtained on one-dimensional periodic domains from the non-linear phase-diffusion equation, the Burgers equation, the Korteweg-de Vries equation, and a coupled FitzHugh-Nagumo system of partial differential equations. The reconstruction capability of tieDMD is compared to existing linear and non-linear variants of the dynamic mode decomposition applied to the same data. For the regarded data, tieDMD consistently shows superior capabilities in data reconstruction.