Numerical Gaussian Process Kalman Filtering for Spatiotemporal Systems

We present a novel Kalman filter (KF) for spatiotemporal systems called the numerical Gaussian process Kalman filter (NGPKF). 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 while also providing uncertainty quantification of the simulation resulting from noisy initial data. We formulate numerical Gaussian processes as linear Gaussian state space models. This allows us to derive the recursive KF algorithm under the numerical Gaussian process state space model. Using two case studies, we show that the NGPKF is more accurate and robust, given enough measurements, than a spatial discretization-based KF.