  Data in many applications follows systems of Ordinary Differential Equations
(ODEs). This paper presents a novel algorithmic and symbolic construction for
covariance functions of Gaussian Processes (GPs) with realizations strictly
following a system of linear homogeneous ODEs with constant coefficients, which
we call LODE-GPs. Introducing this strong inductive bias into a GP improves
modelling of such data. Using smith normal form algorithms, a symbolic
technique, we overcome two current restrictions in the state of the art: (1)
the need for certain uniqueness conditions in the set of solutions, typically
assumed in classical ODE solvers and their probabilistic counterparts, and (2)
the restriction to controllable systems, typically assumed when encoding
differential equations in covariance functions. We show the effectiveness of
LODE-GPs in a number of experiments, for example learning physically
interpretable parameters by maximizing the likelihood.
