Multigrid-augmented deep learning preconditioners for the Helmholtz equation

TL;DR

This paper introduces a novel data-driven preconditioning method combining CNNs with multigrid techniques to efficiently solve high-frequency Helmholtz equations, demonstrating improved robustness and generalization in 2D problems.

Contribution

It proposes a new CNN-based preconditioner integrated with multigrid methods for the Helmholtz equation, including a robust encoder-solver framework and a mini-retraining procedure.

Findings

Effective generalization over residuals and wave models.

Enhanced robustness and efficiency in 2D Helmholtz problems.

Improved performance with mini-retraining for multiple right-hand sides.

Abstract

In this paper, we present a data-driven approach to iteratively solve the discrete heterogeneous Helmholtz equation at high wavenumbers. In our approach, we combine classical iterative solvers with convolutional neural networks (CNNs) to form a preconditioner which is applied within a Krylov solver. For the preconditioner, we use a CNN of type U-Net that operates in conjunction with multigrid ingredients. Two types of preconditioners are proposed 1) U-Net as a coarse grid solver, and 2) U-Net as a deflation operator with shifted Laplacian V-cycles. Following our training scheme and data-augmentation, our CNN preconditioner can generalize over residuals and a relatively general set of wave slowness models. On top of that, we also offer an encoder-solver framework where an "encoder" network generalizes over the medium and sends context vectors to another "solver" network, which generalizes over the right-hand-sides. We show that this option is more robust and efficient than the stand-alone variant. Lastly, we also offer a mini-retraining procedure, to improve the solver after the model is known. This option is beneficial when solving multiple right-hand-sides, like in inverse problems. We demonstrate the efficiency and generalization abilities of our approach on a variety of 2D problems.