Attention-based hybrid solvers for linear equations that are geometry aware

TL;DR

This paper introduces a geometry-aware deep learning architecture that learns preconditioners for linear PDEs, demonstrating robustness across different geometries without retraining, especially for the Helmholtz equation.

Contribution

It presents a novel deep operator network that generalizes preconditioning for PDEs across geometries without additional data or fine-tuning.

Findings

Effective preconditioning for Helmholtz equation demonstrated

Robustness across different geometries without retraining

Applicable to non-positive definite and non-symmetric PDEs

Abstract

We present a novel architecture for learning geometry-aware preconditioners for linear partial differential equations (PDEs). We show that a deep operator network (Deeponet) can be trained on a simple geometry and remain a robust preconditioner for problems defined by different geometries without further fine-tuning or additional data mining. We demonstrate our method for the Helmholtz equation, which is used to solve problems in electromagnetics and acoustics; the Helmholtz equation is not positive definite, and with absorbing boundary conditions, it is not symmetric.