In this article we propose a new deep learning approach to solve parametric partial differential equations (PDEs) approximately. 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 scientific computing approximation problem under consideration. In the proposed approach we combine efficient classical numerical approximation techniques such as higher-order Runge-Kutta schemes with sophisticated deep (operator) learning methodologies such as the recently introduced Fourier neural operators (FNOs). Specifically, we introduce customized adaptions of existing standard 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 approach in the case of some parametric PDEs. In the tested numerical examples the ADANN approach significantly outperforms existing traditional approximation algorithms as well as existing deep learning methodologies from the literature.

翻译：本文提出了一种新的深度学习方法，用于近似求解参数化偏微分方程（PDEs）。具体而言，我们引入了一种新策略，针对特定的科学计算近似问题，设计了特定的人工神经网络（ANN）架构，并搭配专门的ANN初始化方案。该方法将高效的经典数值近似技术（如高阶龙格-库塔格式）与先进的深度（算子）学习方法（如近期提出的傅立叶神经算子（FNOs））相结合。具体地，我们对现有标准ANN架构进行了定制化调整，并为这些架构设计了专门的初始化方式，使得在初始化阶段，ANN能紧密模仿所选定的针对该近似问题的高效经典数值算法。由此获得的ANN架构及其初始化方案深受数值算法及文献中流行的深度学习方法的启发，因此我们将这些结合的ANN及其定制初始化方案称为算法设计的人工神经网络（ADANNs）。我们针对若干参数化PDEs对所提出的ADANN方法进行了数值测试。在测试的数值算例中，ADANN方法显著优于现有的传统逼近算法及文献中的深度学习方法。