Operator approximation of the wave equation based on deep learning of Green's function

TL;DR

This paper introduces GreenONets, a novel deep learning approach leveraging Green's functions to efficiently approximate solutions to the wave equation, demonstrating improved performance over traditional DeepONets in various media.

Contribution

The paper presents GreenONets, a new operator network architecture that incorporates Green's functions for better wave equation approximation, enhancing convergence and accuracy.

Findings

GreenONets outperform DeepONets in wave equation approximation.

GreenONets achieve faster convergence in parameter identification.

Effective in both homogeneous and heterogeneous media.

Abstract

Deep operator networks (DeepONets) have demonstrated their capability of approximating nonlinear operators for initial- and boundary-value problems. One attractive feature of DeepONets is their versatility since they do not rely on prior knowledge about the solution structure of a problem and can thus be directly applied to a large class of problems. However, convergence in identifying the parameters of the networks may sometimes be slow. In order to improve on DeepONets for approximating the wave equation, we introduce the Green operator networks (GreenONets), which use the representation of the exact solution to the homogeneous wave equation in term of the Green's function. The performance of GreenONets and DeepONets is compared on a series of numerical experiments for homogeneous and heterogeneous media in one and two dimensions.