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

Ido Ben-Yair, Bar Lerer, Eran Treister

Keywords: Helmholtz equation, shifted Laplacian, multigrid, iterative methods, deep learning, convolutional neural networks, implicit methods, Lippmann-Schwinger equation

TL;DR: We solve the discrete heterogeneous Helmholtz equation using a deep CNN with a global implicit layer inspired by the Lippmann-Schwinger equation

Abstract:

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.

Submission Number: 86