  A novel multi-level method for partial differential equations with uncertain
parameters is proposed. The principle behind the method is that the error
between grid levels in multi-level methods has a spatial structure that is by
good approximation independent of the actual grid level. Our method learns this
structure by employing a sequence of convolutional neural networks, that are
well-suited to automatically detect local error features as latent quantities
of the solution. Furthermore, by using the concept of transfer learning, the
information of coarse grid levels is reused on fine grid levels in order to
minimize the required number of samples on fine levels. The method outperforms
state-of-the-art multi-level methods, especially in the case when complex PDEs
(such as single-phase and free-surface flow problems) are concerned, or when
high accuracy is required.
