  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.
