  We combine concepts from multilevel solvers for partial differential
equations (PDEs) with neural network based deep learning and propose a new
methodology for the efficient numerical solution of high-dimensional parametric
PDEs. An in-depth theoretical analysis shows that the proposed architecture is
able to approximate multigrid V-cycles to arbitrary precision with the number
of weights only depending logarithmically on the resolution of the finest mesh.
As a consequence, approximation bounds for the solution of parametric PDEs by
neural networks that are independent on the (stochastic) parameter dimension
can be derived. The performance of the proposed method is illustrated on
high-dimensional parametric linear elliptic PDEs that are common benchmark
problems in uncertainty quantification. We find substantial improvements over
state-of-the-art deep learning-based solvers. As particularly challenging
examples, random conductivity with high-dimensional non-affine Gaussian fields
in 100 parameter dimensions and a random cookie problem are examined. Due to
the multilevel structure of our method, the amount of training samples can be
reduced on finer levels, hence significantly lowering the generation time for
training data and the training time of our method.
