  We infer both microscopic and macroscopic behaviors of a three-dimensional
chaotic fluid flow using reservoir computing. In our procedure of the
inference, we assume no prior knowledge of a physical process of a fluid flow
except that its behavior is complex but deterministic. We present two ways of
inference of the complex behavior; the first called partial-inference requires
continued knowledge of partial time-series data during the inference as well as
past time-series data, while the second called full-inference requires only
past time-series data as training data. For the first case, we are able to
infer long-time motion of microscopic fluid variables. For the second case, we
show that the reservoir dynamics constructed from only past data of energy
functions can infer the future behavior of energy functions and reproduce the
energy spectrum. It is also shown that we can infer a time-series data from
only one measurement by using the delay coordinates. These implies that the
obtained two reservoir systems constructed without the knowledge of microscopic
data are equivalent to the dynamical systems describing macroscopic behavior of
energy functions.
