Numerical Gaussian process Kalman filtering for spatiotemporal systems

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.