  This study evaluates data-driven models from a dynamical system perspective,
such as unstable fixed points, periodic orbits, Lyapunov exponents, manifold
structures, and statistical values. These dynamical characteristics can be
reconstructed much more precisely by a data-driven model using reservoir
computing than by computing directly from training data. With this idea, we
estimate the state-lasting time distribution of a particular macroscopic
variable of chaotic fluid flow, which cannot be calculated from a direct
numerical simulation of the Navier--Stokes equation because of its high
computational cost.
