  A statistical learning approach for parametric PDEs related to Uncertainty
Quantification is derived. The method is based on the minimization of an
empirical risk on a selected model class and it is shown to be applicable to a
broad range of problems. A general unified convergence analysis is derived,
which takes into account the approximation and the statistical errors. By this,
a combination of theoretical results from numerical analysis and statistics is
obtained. Numerical experiments illustrate the performance of the method with
the model class of hierarchical tensors.
