  Purely data driven approaches for machine learning present difficulties when
data is scarce relative to the complexity of the model or when the model is
forced to extrapolate. On the other hand, purely mechanistic approaches need to
identify and specify all the interactions in the problem at hand (which may not
be feasible) and still leave the issue of how to parameterize the system. In
this paper, we present a hybrid approach using Gaussian processes and
differential equations to combine data driven modelling with a physical model
of the system. We show how different, physically-inspired, kernel functions can
be developed through sensible, simple, mechanistic assumptions about the
underlying system. The versatility of our approach is illustrated with three
case studies from motion capture, computational biology and geostatistics.
