Learning Rays via Deep Neural Network in a Ray-based IPDG Method for High-Frequency Helmholtz Equations in Inhomogeneous Media

TL;DR

This paper introduces a deep learning method to determine ray directions from wave fields, enhancing high-frequency Helmholtz equation solutions in inhomogeneous media without pollution effects, and demonstrating high accuracy and efficiency.

Contribution

A novel deep neural network approach for extracting ray directions that improves the accuracy of high-frequency Helmholtz solutions in complex media.

Findings

No apparent pollution effects in solutions

High accuracy in 2D and 3D simulations

Efficient computation with deep learning integration

Abstract

We develop a deep learning approach to extract ray directions at discrete locations by analyzing highly oscillatory wave fields. A deep neural network is trained on a set of local plane-wave fields to predict ray directions at discrete locations. The resulting deep neural network is then applied to a reduced-frequency Helmholtz solution to extract the directions, which are further incorporated into a ray-based interior-penalty discontinuous Galerkin (IPDG) method to solve the Helmholtz equations at higher frequencies. In this way, we observe no apparent pollution effects in the resulting Helmholtz solutions in inhomogeneous media. Our 2D and 3D numerical results show that the proposed scheme is very efficient and yields highly accurate solutions.