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.

翻译：本文提出了一种新的深度学习方法来逼近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，用于设计与特定人工神经网络（ANN）初始化方案相结合的专用ANN架构，这些架构是为所考虑的特定逼近问题量身定制的。在所提出的方法中，我们将高效的经典数值逼近技术与深度算子学习方法相结合。具体来说，我们引入了对现有ANN架构的定制化改造以及针对这些ANN架构的专用初始化方案，使得在初始化时，ANN能够紧密模拟针对所考虑逼近问题所选定的高效经典数值算法。因此，所获得的ANN架构及其初始化方案深受数值算法以及文献中流行的深度学习方法的启发，从这个意义上，我们将引入的ANN与其量身定制的初始化方案统称为算法化设计的人工神经网络（ADANNs）。我们在多个参数化PDE的案例中对所提出的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传统逼近算法以及文献中现有的深度算子学习方法。