Wavefield Solutions from Machine Learned Functions that Approximately Satisfy the Wave Equation

Abstract

Solving the Helmholtz equation provides wavefield solutions that are dimensionally compressed, per frequency, compared to the time domain which is useful for many applications, like full waveform inversion (FWI). However, the efficiency in attaining such wavefield solutions depends often on the size of the model, which tends to be large at high frequencies and for 3D problems. Thus, we use here a recently introduced framework based on neural networks to predict such solutions through setting the underlying physical equation as a loss function to optimize the neural network parameters. For an input point in the model space, the network learns to predict the wavefield value at that point, and its partial derivatives using a concept referred to as automatic differentiation, to fit, in our case, a form of Helmholtz equation. We specifically seek the solution of the scattered wavefield considering a simple homogeneous background model that allows for analytical solutions. Feeding a reasonable number of random points from the model space will ultimately train a fully connected 8-layer deep neural network with each layer having a dimension of 20, to predict the scattered wavefield function. Initial tests on a two-box-shaped scatterer model with a source in the middle, as well as, a layered model with a source on the surface demonstrate the successful training of the NN for this application and provide us with a peek into the potential of such an approach.

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

“…For the isotropic case, the outputs are given by the real and imaginary parts of the pressure wavefield at these points; while for the anisotropic case, the outputs are given by the real and imaginary parts of the pressure and auxiliary perturbation wavefields. We borrow the PINN architecture from previous work (Raissi et al 2019a;Alkhalifah et al 2020b;Song et al 2021), which proves to be efficient and effective. In all the experiments shown in this paper, an Adam optimizer and a follow-up L-BFGS optimization, which is a quasi-Newton approach, with full-batch are used to optimize the loss function (Liu & Nocedal 1989).…”

“…Thanks to the explosive growth of the available data and the recent developments in computing resources, NNs have become more and more widely used in geophysics, first arrivals and phases of P-and S-waves picking (Dai & MacBeth 1995;Gentili & Michelini 2006;Zhu & Beroza 2019), automatic normal moveout (NMO) correction (Calderón-Mac ı´as et al 1998), seismograms analysis and quality assessment (Valentine & Trampert 2012), seismic ground-roll noise attenuation (Kaur et al 2020) modify the loss functions corresponding to the underlying physical laws. In geophysical applications, PINNs have shown its effectiveness in time-and frequency-domain wave equation modelling (Karimpouli & Tahmasebi 2020;Alkhalifah et al 2020b), and solving the isotropic and anisotropic P-wave eikonal equations (Smith et al 2020;Waheed et al 2020a,b), and magnetotelluric forward modelling (Wang et al 2021b). Alkhalifah et al (2020b) used PINNs to solve for the scattered pressure wavefield instead of the whole wavefield directly to avoid the point source singularity, and they also extended their work to VTI media (Song et al 2021).…”

“…In geophysical applications, PINNs have shown its effectiveness in time-and frequency-domain wave equation modelling (Karimpouli & Tahmasebi 2020;Alkhalifah et al 2020b), and solving the isotropic and anisotropic P-wave eikonal equations (Smith et al 2020;Waheed et al 2020a,b), and magnetotelluric forward modelling (Wang et al 2021b). Alkhalifah et al (2020b) used PINNs to solve for the scattered pressure wavefield instead of the whole wavefield directly to avoid the point source singularity, and they also extended their work to VTI media (Song et al 2021). Though they show the effectiveness and accuracy of solving for the scattered wave equation using PINN, the resulting scattered wavefields are generally smooth, and fail to describe the detailed information of the subsurface.…”

Wavefield Reconstruction Inversion via Physics-Informed Neural Networks

“…Using the concept of automatic differentiation [56], PINN can easily calculate the partial derivatives of NNs with respect to the input data, which often are spatial and temporal coordinates. In geophysical applications, PINNs have already shown effectiveness in solving the isotropic and anisotropic P-wave eikonal equation [57], [58], Helmholtz equations for isotropic and anisotropic acoustic media [59], [60], [61]. In these applications, the spatial coordinate values are used as input data, and the velocity and anisotropic parameters are considered as implicit parameters in the loss function.…”

“…In generating the frequency-domain wavefields using PINN, [59] proposed to solve the scattered form of the Helmholtz equation to avoid the point source singularity. An infinite isotropic homogeneous model is used as the background model to get analytical solutions of the background wavefield, which is computationally cheap and flexible of the model shapes.…”