  Integro-difference equation (IDE) models describe the conditional dependence
between the spatial process at a future time point and the process at the
present time point through an integral operator. Nonlinearity or temporal
dependence in the dynamics is often captured by allowing the operator
parameters to vary temporally, or by re-fitting a model with a
temporally-invariant linear operator in a sliding window. Both procedures tend
to be excellent for prediction purposes over small time horizons, but are
generally time-consuming and, crucially, do not provide a global prior model
for the temporally-varying dynamics that is realistic. Here, we tackle these
two issues by using a deep convolution neural network (CNN) in a hierarchical
statistical IDE framework, where the CNN is designed to extract process
dynamics from the process' most recent behaviour. Once the CNN is fitted,
probabilistic forecasting can be done extremely quickly online using an
ensemble Kalman filter with no requirement for repeated parameter estimation.
We conduct an experiment where we train the model using 13 years of daily
sea-surface temperature data in the North Atlantic Ocean. Forecasts are seen to
be accurate and calibrated. A key advantage of our approach is that the CNN
provides a global prior model for the dynamics that is realistic,
interpretable, and computationally efficient. We show the versatility of the
approach by successfully producing 10-minute nowcasts of weather radar
reflectivities in Sydney using the same model that was trained on daily
sea-surface temperature data in the North Atlantic Ocean.
