  In this manuscript we introduce numerical Gaussian process Kalman filtering
(GPKF). Numerical Gaussian processes have recently been developed to simulate
spatiotemporal models. The contribution of this paper is to embed numerical
Gaussian processes into the recursive Kalman filter equations. This embedding
enables us to do Kalman filtering on infinite-dimensional systems using
Gaussian processes. This is possible because i) we are obtaining a linear model
from numerical Gaussian processes, and ii) the states of this model are by
definition Gaussian distributed random variables. Convenient properties of the
numerical GPKF are that no spatial discretization of the model is necessary,
and manual setting up of the Kalman filter, that is fine-tuning the process and
measurement noise levels by hand is not required, as they are learned online
from the data stream. We showcase the capability of the numerical GPKF in a
simulation study of the advection equation.
