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，用于求解高频离散非均匀亥姆霍兹方程。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，获得了一个比标准多网格求解器速度更快、扩展性更好的学习型神经求解器。与以往同类神经方法相比，本方法有三项主要创新。第一，构建了多层级U型编解码器结构CNN，并在U型网络的最粗网格层引入隐式层（其卷积核可求逆），从而缓解了CNN的视野受限问题，提升了扩展能力。第二，在参数数量、计算时间和收敛速度方面改进了已有CNN预条件器。第三，提出多尺度训练策略，使网络在保持合理训练流程的同时，能够扩展至未见过维度的问题。所提出的编解码器架构可泛化至不同难度的慢度模型，且能高效求解每个慢度模型对应的多个右端项问题。通过高频非均匀二维问题的数值实验，验证了本新型架构的优势。