  We introduce a data-driven method and shows its skills for spatiotemporal
prediction of high-dimensional chaotic dynamics and turbulence. The method is
based on a finite-dimensional approximation of the Koopman operator where the
observables are vector-valued and delay-embedded, and the nonlinearities are
treated as external forcings. The predictive capabilities of the method are
demonstrated for well-known prototypes of chaos such as the
Kuramoto-Sivashinsky equation and Lorenz-96 system, for which the data-driven
predictions are accurate for several Lyapunov timescales. Similar performance
is seen for two-dimensional lid-driven cavity flows at high Reynolds numbers.
