  In this article, we present data-driven reduced-order modeling for
nonautonomous dynamical systems in multiscale media using Koopman operators.
Different from the case of autonomous dynamical systems, the Koopman operator
family of nonautonomous dynamical systems significantly depend on a time pair.
In order to effectively estimate the time-dependent Koopman operators, a moving
time window is used to decompose the snapshot data, and the extended dynamic
mode decomposition method is applied to computing the Koopman operators in each
local temporal domain. Many physical properties in multiscale media often vary
in very different scales. In order to capture multiscale information well, the
dimension of collected data may be high. To accurately construct the models of
dynamical systems in multiscale media, we use high spatial dimension of
observation data. It is challenging to compute the Koopman operators using the
very high dimensional data. Thus, the strategy of reduced-order modeling is
proposed to treat the difficulty. The proposed reduced-order modeling includes
two stages: offline stage and online stage. In offline stage, a block-wise low
rank decomposition is used to reduce the spatial dimension of initial snapshot
data. For the nonautonomous dynamical systems, real-time observation data may
be required to update the Koopman operators. The online reduced-order modeling
is proposed to correct the offline reduced-order modeling. Three methods are
developed for the online reduced-order modeling: fully online, semi-online and
adaptive online. The adaptive online method automatically selects the fully
online or semi-online and can achieve a good trade-off between modeling
accuracy and efficiency.
