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：我们提出了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齐次亥姆霍兹方程。通过预条件技术将经典迭代多网格求解器与卷积神经网络（CNN）相结合，我们获得了一种学习型神经求解器，其速度优于标准多网格求解器，且具备更好的可扩展性。与以往同类神经方法相比，我们的方法主要贡献有三点。第一，我们构建了一个多级U-Net式编码-求解CNN，并在U-Net最粗网格层引入隐式层，通过反转卷积核来缓解CNN的视场问题，从而实现更好的可扩展性。第二，我们在参数量、计算时间和收敛速度方面改进了原有的CNN预条件子。第三，我们提出了一种多尺度训练方法，使网络能够扩展到先前未见维度的问题，同时保持合理的训练流程。我们的编码-求解架构可泛化至不同难度的多种慢度模型，并能在单个慢度模型下高效求解多个右端项。通过高波数下多种非齐次二维问题的数值实验，我们验证了新型架构的优势。