Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layers

We present a deep learning-based iterative approach to solve the discreteheterogeneous Helmholtz equation for high wavenumbers. Combining classicaliterative multigrid solvers and convolutional neural networks (CNNs) viapreconditioning, we obtain a learned neural solver that is faster and scalesbetter than a standard multigrid solver. Our approach offers three maincontributions over previous neural methods of this kind. First, we construct amultilevel U-Net-like encoder-solver CNN with an implicit layer on the coarsestgrid of the U-Net, where convolution kernels are inverted. This alleviates thefield of view problem in CNNs and allows better scalability. Second, we improveupon the previous CNN preconditioner in terms of the number of parameters,computation time, and convergence rates. Third, we propose a multiscaletraining approach that enables the network to scale to problems of previouslyunseen dimensions while still maintaining a reasonable training procedure. Ourencoder-solver architecture can be used to generalize over different slownessmodels of various difficulties and is efficient at solving for many right-handsides per slowness model. We demonstrate the benefits of our novel architecturewith numerical experiments on a variety of heterogeneous two-dimensionalproblems at high wavenumbers.