Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation Using Compact Implicit Layers
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.
Cite
Text
Ben-Yair et al. "Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation Using Compact Implicit Layers." ICLR 2024 Workshops: AI4DiffEqtnsInSci, 2024.Markdown
[Ben-Yair et al. "Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation Using Compact Implicit Layers." ICLR 2024 Workshops: AI4DiffEqtnsInSci, 2024.](https://mlanthology.org/iclrw/2024/benyair2024iclrw-multigridaugmented/)BibTeX
@inproceedings{benyair2024iclrw-multigridaugmented,
title = {{Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation Using Compact Implicit Layers}},
author = {Ben-Yair, Ido and Lerer, Bar and Treister, Eran},
booktitle = {ICLR 2024 Workshops: AI4DiffEqtnsInSci},
year = {2024},
url = {https://mlanthology.org/iclrw/2024/benyair2024iclrw-multigridaugmented/}
}