Anisotropic eikonal solution using physics-informed neural networks

Waheed, Umair bin; Haghighat, Ehsan; Alkhalifah, Tariq

doi:10.1190/segam2020-3423159.1

Cited by 9 publications

(5 citation statements)

References 17 publications

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“…Another application of the civil engineering field is seismic wave propagation, where time domain wave equations are computationally expensive as they need a lot of memory to store the wavefield solutions. With this in view, Song et al [36]. To solve the scattered pressure wavefield, Alkhalifah et al [34] proposed a PINN-based framework.…”

Section: Surrogate Modelmentioning

confidence: 99%

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State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

Das¹,

Tesfamariam²

2022

Preprint

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This paper presents a comprehensive review of the design of experiments used in the surrogate models. In particular, this study demonstrates the necessity of the design of experiment schemes for the Physics-Informed Neural Network (PINN), which belongs to the supervised learning class. Many complex partial differential equations (PDEs) do not have any analytical solution; only numerical methods are used to solve the equations, which is computationally expensive. In recent decades, PINN has gained popularity as a replacement for numerical methods to reduce the computational budget. PINN uses physical information in the form of differential equations to enhance the performance of the neural networks. Though it works efficiently, the choice of the design of experiment scheme is important as the accuracy of the predicted responses using PINN depends on the training data. In this study, five different PDEs are used for numerical purposes, i.e., viscous Burger's equation, Shrödinger equation, heat equation, Allen-Cahn equation, and Korteweg-de Vries equation. A comparative study is performed to establish the necessity of the selection of a DoE scheme. It is seen that the Hammersley sampling-based PINN performs better than other DoE sample strategies.

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Section: Surrogate Modelmentioning

confidence: 99%

“…22. The boundary conditions are expressed as u(0, t) = 0 u(L, t) = 1 (36) In this study, the parameters L and T 0 are taken as 1 and 0.5, respectively. The optimal neural network parameters are estimated by minimizing the loss function L = L u + L f .…”

Section: Problem 3: 1d -Heat Equationmentioning

confidence: 99%

State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

Das¹,

Tesfamariam²

2022

Preprint

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“…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).…”

Section: Introductionmentioning

confidence: 99%

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

Chen,

de Ridder,

Rost

et al. 2023

JGR Solid Earth

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We develop a novel approach for multi‐frequency, elliptical‐anisotropic eikonal tomography based on physics‐informed neural networks (pinnEAET). This approach simultaneously estimates the medium properties controlling anisotropic Rayleigh waves and reconstructs the traveltimes. The physics constraints built into pinnEAET's neural network enable high‐resolution results with limited inputs by inferring physically plausible models between data points. Even with a single source, pinnEAET can achieve stable convergence on key features where traditional methods lack resolution. We apply pinnEAET to ambient noise data from a dense seismic array (ChinArray‐Himalaya II) in the northeastern Tibetan Plateau with only 20 quasi‐randomly distributed stations as sources. Anisotropic phase velocity maps for Rayleigh waves in the period range from 10–40 s are obtained by training on observed traveltimes. Despite using only about 3% of the total stations as sources, our results show low uncertainties, good resolution and are consistent with results from conventional tomography.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Wavefield Reconstruction Inversion via Physics-Informed Neural Networks

Song

Alkhalifah

2022

IEEE Trans. Geosci. Remote Sensing

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Wavefield reconstruction inversion (WRI) formulates a PDE-constrained optimization problem to reduce cycle skipping in full-waveform inversion (FWI). WRI is often implemented by solving for the frequency-domain representation of the wavefield using the finite-difference method. The approach requires matrix inversions and affords limited flexibility to accommodating irregular model geometries. On the other hand, physics-informed neural network (PINN) uses the underlying physical laws as loss functions to train the neural network (NN) to provide flexible continuous functional approximations of the solutions without matrix inversions. By including a dataconstrained term in the loss function, the trained NN can reconstruct a wavefield that simultaneously fits the recorded data and satisfies the Helmholtz equation for a given initial velocity model. Using the predicted wavefields, we rely on a small-size NN to predict the velocity using the reconstructed wavefield. In this velocity prediction NN, spatial coordinates are used as input data to the network and the scattered Helmholtz equation is used to define the loss function. After we train this network, we are able to predict the velocity in the domain of interest. We develop this PINN-based WRI method and demonstrate its potential using a part of the Sigsbee2A model and a modified Marmousi model. The results show that the PINN-based WRI is able to invert for a reasonable velocity with very limited iterations and frequencies, which can be used in a subsequent FWI application.

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Anisotropic eikonal solution using physics-informed neural networks

Cited by 9 publications

References 17 publications

State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

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

Wavefield Reconstruction Inversion via Physics-Informed Neural Networks

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