Solving the frequency-domain acoustic VTI wave equation using physics-informed neural networks

Chao Song, Tariq Ali Alkhalifah, Umair Bin Waheed

Research output: Contribution to journalArticlepeer-review

107 Scopus citations


SUMMARY Frequency-domain wavefield solutions corresponding to the anisotropic acoustic wave equation can be used to describe the anisotropic nature of the Earth. To solve a frequency-domain wave equation, we often need to invert the impedance matrix. This results in a dramatic increase in computational cost as the model size increases. It is even a bigger challenge for anisotropic media, where the impedance matrix is far more complex. In addition, the conventional finite-difference method produces numerical dispersion artefacts in solving acoustic wave equations for anisotropic media. To address these issues, we use the emerging paradigm of physics-informed neural networks (PINNs) to obtain wavefield solutions for an acoustic wave equation for transversely isotropic (TI) media with a vertical axis of symmetry (VTI). PINNs utilize the concept of automatic differentiation to calculate their partial derivatives, which are free of numerical dispersion artefacts. Thus, we use the wave equation as a loss function to train a neural network to provide functional solutions to the acoustic VTI form of the wave equation. Instead of predicting the pressure wavefields directly, we solve for the scattered pressure wavefields to avoid dealing with the point-source singularity. We use the spatial coordinates as input data to the network, which outputs the real and imaginary parts of the scattered wavefields and auxiliary function. After training a deep neural network, we can evaluate the wavefield at any point in space almost instantly using this trained neural network without calculating the impedance matrix inverse. We demonstrate these features on a simple 2-D anomaly model and a 2-D layered model. Additional tests on a modified 3-D Overthrust model and a 2-D model with irregular topography further validate the effectiveness of the proposed method.
Original languageEnglish (US)
Pages (from-to)846-859
Number of pages14
JournalGeophysical Journal International
Issue number2
StatePublished - Jan 11 2021

Bibliographical note

KAUST Repository Item: Exported on 2021-04-27
Acknowledgements: We thank KAUST for its support and the SWAG group for the collaborative environment. This work utilized the resources of the Supercomputing Laboratory at King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia, and we are grateful for that. We thank the editor, Dr Andrew Valentine, assistant editor, Fern Storey, and two reviewers Dr Anandaroop Ray and Dr Tarje Nissen-Meyer, for their critical and helpful review of the manuscript. We thank Bin She, from the University of Electronic Science and Technology, for sharing the 3-D plotting tool. We thank Dr Fabio Crameri for releasing a perceptually uniform colour map.

ASJC Scopus subject areas

  • Geochemistry and Petrology
  • Geophysics


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