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

We present a deep learning-based iterative approach to solve the discrete heterogeneous Helmholtz equation for high wavenumbers.Combining classical iterative multigrid solvers and neural networks via preconditioning, we obtain a faster, learned neural solver that scales better than a standard multigrid solver.We construct a multilevel U-Net-like encoder-solver CNN with an implicit layer on the coarsest level, where convolution kernels are inverted.This alleviates the field of view problem in CNNs and allows better scalability.Furthermore, we propose a multiscale training approach that enables to scale to problems of previously unseen dimensions while still maintaining a reasonable training procedure.