  There is growing interest in data-driven weather prediction (DDWP), for
example using convolutional neural networks such as U-NETs that are trained on
data from models or reanalysis. Here, we propose 3 components to integrate with
commonly used DDWP models in order to improve their physical consistency and
forecast accuracy. These components are 1) a deep spatial transformer added to
the latent space of the U-NETs to preserve a property called equivariance,
which is related to correctly capturing rotations and scalings of features in
spatio-temporal data, 2) a data-assimilation (DA) algorithm to ingest noisy
observations and improve the initial conditions for next forecasts, and 3) a
multi-time-step algorithm, which combines forecasts from DDWP models with
different time steps through DA, improving the accuracy of forecasts at short
intervals. To show the benefit/feasibility of each component, we use
geopotential height at 500~hPa (Z500) from ERA5 reanalysis and examine the
short-term forecast accuracy of specific setups of the DDWP framework. Results
show that the equivariance-preserving networks (U-STNs) clearly outperform the
U-NETs, for example improving the forecast skill by $45\%$. Using a sigma-point
ensemble Kalman (SPEnKF) algorithm for DA and U-STN as the forward model, we
show that stable, accurate DA cycles are achieved even with high observation
noise. The DDWP+DA framework substantially benefits from large ($O(1000)$)
ensembles that are inexpensively generated with the data-driven forward model
in each DA cycle. The multi-time-step DDWP+DA framework also shows promises,
e.g., it reduces the average error by factors of 2-3.
