We present a deep learning-based iterative approach to solve the discrete heterogeneous Helmholtz equation for high wavenumbers. Combining classical iterative multigrid solvers and convolutional neural networks (CNNs) via preconditioning, we obtain a learned neural solver that is faster and scales better than a standard multigrid solver. Our approach offers three main contributions over previous neural methods of this kind. First, we construct a multilevel U-Net-like encoder-solver CNN with an implicit layer on the coarsest grid of the U-Net, where convolution kernels are inverted. This alleviates the field of view problem in CNNs and allows better scalability. Second, we improve upon the previous CNN preconditioner in terms of the number of parameters, computation time, and convergence rates. Third, we propose a multiscale training approach that enables the network to scale to problems of previously unseen dimensions while still maintaining a reasonable training procedure. Our encoder-solver architecture can be used to generalize over different slowness models of various difficulties and is efficient at solving for many right-hand sides per slowness model. We demonstrate the benefits of our novel architecture with numerical experiments on a variety of heterogeneous two-dimensional problems at high wavenumbers.

翻译：我们提出一种基于深度学习的迭代方法，用于求解高波数下的离散非均匀亥姆霍兹方程。通过预处理技术将经典迭代多重网格求解器与卷积神经网络（CNN）相结合，我们获得了一个学习型神经求解器，其求解速度更快且可扩展性优于标准多重网格求解器。与先前同类神经方法相比，本方法具有三项主要贡献：第一，构建了一个多级U-Net结构的编码器-求解器CNN，在U-Net最粗网格层引入隐层，通过反演该层卷积核有效缓解了CNN的视野受限问题，并提升了可扩展性；第二，在参数量、计算时间和收敛速度方面均改进了先前的CNN预条件子；第三，提出一种多尺度训练方法，使网络能够在保持合理训练流程的同时，扩展到未见维度的问题。我们的编码器-求解器架构可适用于不同难度的多种慢度模型，并高效解决每个慢度模型对应多个右端项的问题。通过针对多种高波数二维非均匀问题的数值实验，我们验证了该新型架构的优势。