  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 from physics, geomathematics and control, among them the full
inhomogeneous system of Maxwell's equations. By bringing together stochastic
learning and computer algebra in a novel way, we combine noisy observations
with precise algebraic computations.
