A versatile framework to solve the Helmholtz equation using physics-informed neural networks

Abstract

Summary Solving the wave equation to obtain wavefield solutions is an essential step in illuminating the subsurface using seismic imaging and waveform inversion methods. Here, we utilize a recently introduced machine-learning based framework called physics-informed neural networks (PINNs) to solve the frequency-domain wave equation, which is also referred to as the Helmholtz equation, for isotropic and anisotropic media. Like functions, PINNs are formed by using a fully-connected neural network (NN) to provide the wavefield solution at spatial points in the domain of interest, in which the coordinates of the point form the input to the network. We train such a network by back propagating the misfit in the wave equation for the output wavefield values and their derivatives for many points in the model space. Generally, a hyperbolic tangent activation is used with PINNs, however, we use an adaptive sinusoidal activation function to optimize the training process. Numerical results show that PINNs with adaptive sinusoidal activation functions are able to generate frequency-domain wavefield solutions that satisfy wave equations. We also show the flexibility and versatility of the proposed method for various media, including anisotropy, and for models with strong irregular topography.

Direct Imaging Using Physics Informed Neural Networks

“…We also use a sine activation function in every layer other than the last layer, which is linear. The sine activation function has favorable features for wavefield representation [21,22]. The NN functional provides a continuous representation of the wavefield and the image, as opposed to grid based representations, and such a continuous representation offers many benefits.…”

Physics‐Informed Neural Networks for Elliptical‐Anisotropy Eikonal Tomography: Application to Data From the Northeastern Tibetan Plateau

“…Significantly, PINNs have already shown great potential in seismological applications. For forward problems, PINNs have been applied to the eikonal equation for traveltime calculation in isotropic and anisotropic media (Smith et al., 2020; Taufik et al., 2022; Waheed, Alkhalifah, et al., 2021; Waheed et al., 2020; Waheed, Haghighat, et al., 2021) and directly simulate wave equation solutions for acoustic and elastic wave propagation (Alkhalifah et al., 2020; Karimpouli & Tahmasebi, 2020; Moseley, Markham, & Nissen‐Meyer, 2020; Moseley, Nissen‐Meyer, & Markham, 2020; Song & Wang, 2023; Song et al., 2021, 2022). For inverse problems, PINNs have been proposed for exploration‐scale seismic tomography with the factored eikonal equation (Gou et al., 2022; Waheed, Alkhalifah, et al., 2021; Waheed, Haghighat, et al., 2021) and wavefield reconstruction inversion (Song & Alkhalifah, 2021).…”

Identification of unknown space-varying wavenumber in Helmholtz equation using physics-informed neural networks

“…Song et al 6 solved the frequency-domain anisotropic acoustic wave equation with PINNs. Recently, an adaptive framework 22 for solving the Helmholtz equation is proposed, and the adaptive activation function is introduced to optimize the training process. In this study, we pay attention to the problem of inferring the space-varying wavenumber from the Helmholtz equation.…”