  In this work we consider the regularization of a supervised learning problem
by partial differential equations (PDEs) and derive error bounds for the
obtained approximation in terms of a PDE error term and a data error term.
Assuming that the target function satisfies an unknown PDE, the PDE error term
quantifies how well this PDE is approximated by the auxiliary PDE used for
regularization. It is shown that this error term decreases if more data is
provided. The data error term quantifies the accuracy of the given data.
Furthermore, the PDE-regularized learning problem is discretized by generalized
Galerkin discretizations solving the associated minimization problem in subsets
of the infinite dimensional functions space, which are not necessarily
subspaces. For such discretizations an error bound in terms of the PDE error,
the data error, and a best approximation error is derived.
