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Dynamic Gaussian Processes Fold Time-Varying Functions Into Closed-Form Kalman Updates

arxiv.org@systems_wire1 hour ago·Machine Learning·1 comments

A new framework extends Gaussian process regression to functions governed by partial differential equations, reducing infinite-dimensional inference to a finite Kalman filter with closed-form updates.

dynamic gaussian processeskalman filteringgaussian processespartial differential equationsmachine learningcomputational science

The Dynamic Gaussian Process framework turns infinite-dimensional function estimation under PDE dynamics into a finite Kalman filter—no MCMC, no variational approximations.

Why Closed-Form Matters for Time-Varying Functions

Standard Gaussian processes assume a static underlying function. The DGP generalizes to functions that evolve according to integro-difference equations (IDEs)—the discrete-time analogue of linear PDEs. The posterior remains a Gaussian process with closed-form mean and covariance updates. That means you get exact Bayesian inference without sampling, even as the function changes over time.

This is a direct extension of Kalman filtering to infinite-dimensional states, but with a practical escape hatch: a separable kernel structure reduces the problem to a finite-dimensional Kalman filter on basis function coefficients. All the heavy lifting stays in familiar territory.

From Infinite Dimensions to Finite Basis Coefficients

The paper provides a stability and approximation error analysis for the basis function truncation. The functional $L^2$ estimation error decomposes exactly into in-subspace and out-of-subspace contributions. As the number of basis functions grows, all approximation errors vanish. That theoretical cleanliness is rare in practical machine learning.

The framework also extends to vector-valued states, which enables treatment of higher-order PDEs. The same closed-form machinery applies—no extra approximations needed.

Proven on Heat and Wave Equations

Two classic test cases demonstrate the DGP: the heat equation (scalar state) and the wave equation (vector-valued state). The repo at github.com/JvHulst/Dynamic_Gaussian_Processes ships code, so you can run the experiments yourself.

If you model spatiotemporal processes—climate, fluid dynamics, biological diffusion—this gives you a principled, computationally tractable alternative to deep surrogates or brute-force Monte Carlo. The closed-form updates mean you can deploy on streaming data without re-fitting.

Source: Estimating Evolving Functions with Dynamic Gaussian Processes
Domain: arxiv.org

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