Algorithmic Linearly Constrained Gaussian Processes

We algorithmically construct multi-output Gaussian process priors which satisfy linear differential equations. Our approach attempts to parametrize all solutions of the equations using Gr\"obner bases. If successful, a push forward Gaussian process along the paramerization is the desired prior. We consider several examples, among them the full inhomogeneous system of Maxwell's equations. By bringing together stochastic learning and computeralgebra in a novel way, we combine noisy observations with precise algebraic computations.