  The reconstruction and prediction of full-state flows from sparse data are of
great scientific and engineering significance yet remain challenging,
especially in applications where data are sparse and/or subjected to noise. To
this end, this study proposes a deep-learning assisted non-intrusive reduced
order model (named DCDMD) for high-dimensional flow prediction from sparse
data. Based on the compressed sensing (CS)-Dynamic Mode Decomposition (DMD),
the DCDMD model is distinguished by two novelties. Firstly, a sparse matrix is
defined to overcome the strict random distribution condition of sensor
locations in CS, thus allowing flexible sensor deployments and requiring very
few sensors. Secondly, a deep-learning-based proxy is invoked to acquire
coherent flow modes from the sparse data of high-dimensional flows, thereby
addressing the issue of defining sparsity and the stringent incoherence
condition in the conventional CSDMD. The two advantageous features, combined
with the fact that the model retains flow physics in the online stage, lead to
significant enhancements in accuracy and efficiency, as well as superior
insensitivity to data noises (i.e., robustness), in both reconstruction and
prediction of full-state flows. These are demonstrated by three benchmark
examples, i.e., cylinder wake, weekly-mean sea surface temperature and
isotropic turbulence in a periodic square area.
