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

Understand

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.