Multigrid-augmented deep learning preconditioners for the Helmholtz equation
This addresses computational challenges in wave propagation simulations for fields like geophysics or acoustics, but it is incremental as it builds on existing multigrid and deep learning methods.
The paper tackles solving the discrete heterogeneous Helmholtz equation at high wavenumbers by combining classical iterative solvers with convolutional neural networks (CNNs) to form preconditioners, showing that the approach is efficient and generalizes well on 2D problems.
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.