Numerical Gaussian process Kalman filtering for spatiotemporal systems

TL;DR

This paper introduces the numerical Gaussian process Kalman filter (GPKF), a new probabilistic filtering method for spatiotemporal systems that leverages physics-informed Gaussian processes to model PDEs without spatial discretization.

Contribution

The paper develops a novel Kalman filtering approach by integrating numerical Gaussian processes into a state space framework for the first time.

Findings

Successfully applied to two case studies.

Demonstrates effective modeling of PDEs without spatial discretization.

Provides a probabilistic framework for spatiotemporal filtering.

Abstract

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.