  We present a novel Kalman filter for spatiotemporal systems called the
numerical Gaussian process Kalman filter (GPKF). Numerical Gaussian processes
have recently been introduced as a physics informed machine learning method for
simulating time-dependent partial differential equations without the need for
spatial discretization. We bring numerical GPs into probabilistic state space
form. This model is linear and its states are Gaussian distributed. These
properties enable us to embed the numerical GP state space model into the
recursive Kalman filter algorithm. We showcase the method using two case
studies.
