Dynamic Gaussian Process framework estimates evolving functions from PDEs

A new framework called the Dynamic Gaussian Process (DGP), developed by van Hulst, Heemels, and Antunes, addresses the challenge of estimating functions that evolve over time according to known physical laws. The DGP models continuous functions governed by integro-difference equations (IDEs), which naturally arise from time-discretizing linear partial differential equations (PDEs) like the heat and wave equations. The key innovation is that the DGP extends Gaussian process (GP) regression to handle time-varying functions, while also extending Kalman filtering to infinite-dimensional state spaces. The posterior distribution remains a Gaussian process with closed-form recursive updates for its mean and covariance, avoiding expensive numerical approximations. By leveraging a separable kernel structure, the infinite-dimensional problem is reduced to a finite-dimensional Kalman filter on the coefficients of basis functions, making computation tractable.

The framework further extends to vector-valued states, enabling the treatment of higher-order PDEs. A rigorous stability and approximation error analysis shows that 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. Practical demonstrations on the heat equation (scalar state) and wave equation (vector state) confirm the method's effectiveness. The DGP offers a principled, scalable approach for real-time estimation of evolving systems, with strong theoretical guarantees—particularly valuable for control systems, physics simulation, and engineering applications where underlying dynamics are partially known. Code is available on GitHub.