算法设计的人工神经网络（ADANNs）：用于参数偏微分方程的高阶深度算子学习
Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations
AI总结：
本文提出了一种新的深度学习方法，用于近似参数偏微分方程相关算子，通过定制神经网络架构和初始化方案，结合经典数值方法与深度算子学习，显著优于传统方法。
摘要：
本文提出了一种新的深度学习方法，用于近似参数偏微分方程（PDEs）相关算子。具体而言，我们介绍了一种新的策略，用于设计特定的人工神经网络（ANN）架构及其特定的初始化方案，这些方案是针对特定的近似问题量身定制的。在所提出的方法中，我们结合了高效的经典数值近似技术与深度算子学习方法。具体来说，我们引入了现有ANN架构的定制化改编以及这些ANN架构的专用初始化方案，使得在初始化时，ANNs能够紧密模仿所考虑的近似问题中选择的高效经典数值算法。所获得的ANN架构及其初始化方案因此强烈受到数值算法以及文献中流行深度学习方法的启发。因此，我们称所引入的ANN及其专用初始化方案为算法设计的人工神经网络（ADANNs）。我们在多个参数PDEs的情况下对所提出的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的现有深度算子学习方法。
英文摘要：
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.
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