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

TL;DR

It is shown 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.

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.

Figures (4)

• Figure 1: Geometries used for training
• Figure 2: Tested domains $\Omega$ used in Tables \ref{['tab:exp']}, \ref{['tab:exp2']} and \ref{['tab:expgmews']} for solving \ref{['eq:helm']}.
• Figure 3: Tested exterior domains used in Table \ref{['tab:expobs']} for solving $\Delta u+k^2 u=f$. The boundary conditions on the inner boundary are u=0. The boundary conditions on the outer boundary are $\frac{\partial u}{\partial \vec{\nu}}+\sqrt{-1}k u=0$.
• Figure 4: Tested exterior domains used in Table \ref{['tab:expmoreobs']} solving $\Delta u+k^2 u=f$. The boundary conditions on the inner boundaries are u=0. The boundary conditions on the outer boundary are $\frac{\partial u}{\partial \vec{\nu}}+\sqrt{-1}k u=0$.

Theorems & Definitions (4)

• Remark 1.1
• Remark 2.1
• Remark 3.1
• Remark 3.2