  In this article we propose a new deep learning approach to approximate
operators related to parametric partial differential equations (PDEs). In
particular, we introduce a new strategy to design specific artificial neural
network (ANN) architectures in conjunction with specific ANN initialization
schemes which are tailor-made for the particular approximation problem under
consideration. In the proposed approach we combine efficient classical
numerical approximation techniques with deep operator learning methodologies.
Specifically, we introduce customized adaptions of existing ANN architectures
together with specialized initializations for these ANN architectures so that
at initialization we have that the ANNs closely mimic a chosen efficient
classical numerical algorithm for the considered approximation problem. The
obtained ANN architectures and their initialization schemes are thus strongly
inspired by numerical algorithms as well as by popular deep learning
methodologies from the literature and in that sense we refer to the introduced
ANNs in conjunction with their tailor-made initialization schemes as
Algorithmically Designed Artificial Neural Networks (ADANNs). We numerically
test the proposed ADANN methodology in the case of several parametric PDEs. In
the tested numerical examples the ADANN methodology significantly outperforms
existing traditional approximation algorithms as well as existing deep operator
learning methodologies from the literature.
