  Variational Data Assimilation (DA) has been broadly used in engineering
problems for field reconstruction and prediction by performing a weighted
combination of multiple sources of noisy data. In recent years, the integration
of deep learning (DL) techniques in DA has shown promise in improving the
efficiency and accuracy in high-dimensional dynamical systems. Nevertheless,
existing deep DA approaches face difficulties in dealing with unstructured
observation data, especially when the placement and number of sensors are
dynamic over time. We introduce a novel variational DA scheme, named
Voronoi-tessellation Inverse operator for VariatIonal Data assimilation
(VIVID), that incorporates a DL inverse operator into the assimilation
objective function. By leveraging the capabilities of the Voronoi-tessellation
and convolutional neural networks, VIVID is adept at handling sparse,
unstructured, and time-varying sensor data. Furthermore, the incorporation of
the DL inverse operator establishes a direct link between observation and state
space, leading to a reduction in the number of minimization steps required for
DA. Additionally, VIVID can be seamlessly integrated with Proper Orthogonal
Decomposition (POD) to develop an end-to-end reduced-order DA scheme, which can
further expedite field reconstruction. Numerical experiments in a fluid
dynamics system demonstrate that VIVID can significantly outperform existing DA
and DL algorithms. The robustness of VIVID is also accessed through the
application of various levels of prior error, the utilization of varying
numbers of sensors, and the misspecification of error covariance in DA.
