Estimating Evolving Functions with Dynamic Gaussian Processes
For researchers in spatiotemporal modeling and PDE estimation, this provides a principled Bayesian framework with theoretical guarantees, though it is an incremental extension of existing Gaussian process and Kalman filter methods.
This paper introduces the Dynamic Gaussian Process (DGP) for estimating functions governed by integro-difference equations, extending Gaussian process regression to time-varying functions and Kalman filtering to infinite-dimensional states. The DGP achieves closed-form posterior updates and is demonstrated on the heat and wave equations.
This paper develops the Dynamic Gaussian Process (DGP), a framework for estimating functions governed by integro-difference equations (IDEs). IDEs model continuous functions that evolve with discrete-time dynamics and arise naturally from time-discretization of linear partial differential equations (PDEs). The DGP extends Gaussian process regression to time-varying functions and extends Kalman filtering to infinite-dimensional states. The DGP posterior remains a Gaussian process with closed-form mean and covariance updates, and separable kernel structure reduces the problem to a finite-dimensional Kalman filter on basis function coefficients. This paper extends the DGP to vector-valued states, enabling the treatment of higher-order PDEs, and provides a stability and approximation error analysis for the basis function approximation. The functional L2 estimation error decomposes exactly into in-subspace and out-of-subspace contributions, and all approximation errors vanish as the number of basis functions grows. The framework is demonstrated on the heat equation and on the wave equation, the latter with a vector-valued state. Code is available at https://github.com/JvHulst/Dynamic_Gaussian_Processes.