Local Randomized Neural Networks with Discontinuous Galerkin Methods for Diffusive-Viscous Wave Equation

TL;DR

This paper develops a novel LRNN-DG numerical method for the diffusive-viscous wave equation, demonstrating improved accuracy and efficiency over traditional methods in geophysical applications.

Contribution

It extends LRNN-DG methods to the diffusive-viscous wave equation, providing a new high-accuracy, cost-effective approach for complex wave phenomena.

Findings

Achieves higher accuracy than traditional methods

Reduces computational costs significantly

Handles time-dependent problems efficiently

Abstract

The diffusive-viscous wave equation is an advancement in wave equation theory, as it accounts for both diffusion and viscosity effects. This has a wide range of applications in geophysics, such as the attenuation of seismic waves in fluid-saturated solids and frequency-dependent phenomena in porous media. Therefore, the development of an efficient numerical method for the equation is of both theoretical and practical importance. Recently, local randomized neural networks with discontinuous Galerkin (LRNN-DG) methods have been introduced in \cite{Sun2022lrnndg} to solve elliptic and parabolic equations. Numerical examples suggest that LRNN-DG can achieve high accuracy, and can handle time-dependent problems naturally and efficiently by using a space-time framework. In this paper, we develop LRNN-DG methods for solving the diffusive-viscous wave equation and present numerical experiments with several cases. The numerical results show that the proposed methods can solve the diffusive-viscous wave equation more accurately with less computing costs than traditional methods.