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.

翻译：扩散-黏性波动方程是波动方程理论的重要进展，它同时考虑了扩散和黏性效应。该方程在地球物理学中具有广泛应用，例如流体饱和固体中地震波的衰减以及多孔介质中的频率相关现象。因此，发展该方程的高效数值方法具有重要的理论与实际意义。近期，文献\cite{Sun2022lrnndg}提出了结合间断伽辽金方法的局部随机神经网络（LRNN-DG）方法，用于求解椭圆和抛物型方程。数值算例表明，LRNN-DG方法能够实现高精度，并通过时空框架自然高效地处理时间依赖问题。本文进一步发展了LRNN-DG方法以求解扩散-黏性波动方程，并给出了多个算例的数值实验。数值结果表明，与传统方法相比，所提方法能够以更低的计算成本更精确地求解扩散-黏性波动方程。