  We develop a framework for Gaussian processes regression constrained by
boundary value problems. The framework may be applied to infer the solution of
a well-posed boundary value problem with a known second-order differential
operator and boundary conditions, but for which only scattered observations of
the source term are available. Scattered observations of the solution may also
be used in the regression. The framework combines co-kriging with the linear
transformation of a Gaussian process together with the use of kernels given by
spectral expansions in eigenfunctions of the boundary value problem. Thus, it
benefits from a reduced-rank property of covariance matrices. We demonstrate
that the resulting framework yields more accurate and stable solution inference
as compared to physics-informed Gaussian process regression without boundary
condition constraints.
