Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problem

TL;DR

This paper introduces neural operator-based meta-solvers that efficiently solve 3D scattering problems with complex shapes by leveraging training on non-scattering problems, enabling rapid, shape-agnostic predictions in real-time applications.

Contribution

The paper presents a novel approach combining DeepONet with relaxation and Krylov methods to create shape-agnostic, fast meta-solvers for 3D scattering problems without additional training.

Findings

Meta-solvers are shape-agnostic, fast, and robust.

DeepONet-Krylov methods require significantly fewer iterations.

Standard solvers may fail to converge without DeepONet.

Abstract

Three-dimensional target identification using scattering techniques requires high accuracy solutions and very fast computations for real-time predictions in some critical applications. We first train a deep neural operator~(DeepONet) to solve wave propagation problems described by the Helmholtz equation in a domain \textit{without scatterers} but at different wavenumbers and with a complex absorbing boundary condition. We then design two classes of fast meta-solvers by combining DeepONet with either relaxation methods, such as Jacobi and Gauss-Seidel, or with Krylov methods, such as GMRES and BiCGStab, using the trunk basis of DeepONet as a coarse-scale preconditioner. We leverage the spectral bias of neural networks to account for the lower part of the spectrum in the error distribution while the upper part is handled inexpensively using relaxation methods or fine-scale preconditioners. The meta-solvers are then applied to solve scattering problems with different shape of scatterers, at no extra training cost. We first demonstrate that the resulting meta-solvers are shape-agnostic, fast, and robust, whereas the standard standalone solvers may even fail to converge without the DeepONet. We then apply both classes of meta-solvers to scattering from a submarine, a complex three-dimensional problem. We achieve very fast solutions, especially with the DeepONet-Krylov methods, which require orders of magnitude fewer iterations than any of the standalone solvers.