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 classical approximation algorithms as well as existing deep operator learning methodologies from the literature.

翻译：本文提出一种新的深度学习方法，用于近似与参数化偏微分方程（PDEs）相关的算子。具体而言，我们引入了一种新策略来设计特定的人工神经网络（ANN）架构，并结合针对特定近似问题量身定制的ANN初始化方案。在所提出的方法中，我们将高效的经典数值近似技术深度算子学习方法相结合。具体而言，我们对现有ANN架构进行定制化调整，并为这些架构设计专用初始化方案，使得在初始化时ANN能够紧密模仿针对所考虑近似问题选定的高效经典数值算法。所获得的ANN架构及其初始化方案深受数值算法和文献中流行深度学习方法的启发，因此我们将这些引入的ANN及其定制化初始化方案统称为算法设计人工神经网络（ADANNs）。我们针对多个参数化PDEs对所提出的ADANN方法进行了数值测试。在测试的数值示例中，ADANN方法显著优于现有的经典近似算法以及文献中已有的深度算子学习方法。