Adaptive Multilevel Neural Networks for Parametric PDEs with Error
  Estimation
Janina E. Schütte Martin Eigel
To solve high-dimensional parameter-dependent partial differential equations
(pPDEs), a neural network architecture is presented. It is constructed to map
parameters of the model data to corresponding finite element solutions. To
improve training efficiency and to enable control of the approximation error,
the network mimics an adaptive finite element method (AFEM). It outputs a
coarse grid solution and a series of corrections as produced in an AFEM,
allowing a tracking of the error decay over successive layers of the network.
The observed errors are measured by a reliable residual based a posteriori
error estimator, enabling the reduction to only few parameters for the
approximation in the output of the network. This leads to a problem adapted
representation of the solution on locally refined grids. Furthermore, each
solution of the AFEM is discretized in a hierarchical basis. For the
architecture, convolutional neural networks (CNNs) are chosen. The hierarchical
basis then allows to handle sparse images for finely discretized meshes.
Additionally, as corrections on finer levels decrease in amplitude, i.e.,
importance for the overall approximation, the accuracy of the network
approximation is allowed to decrease successively. This can either be
incorporated in the number of generated high fidelity samples used for training
or the size of the network components responsible for the fine grid outputs.
The architecture is described and preliminary numerical examples are presented.