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

Chao Song, Tariq Ali Alkhalifah, Umair bin Waheed

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


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.
Original languageEnglish (US)
JournalGeophysical Journal International
StatePublished - Oct 23 2021

Bibliographical note

KAUST Repository Item: Exported on 2021-10-26
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, Louise Alexander, and Dr. Martijn van den Ende and one anonymous reviewer, for their critical and helpful review of the manuscript. We thank Dr. Fabio Crameri for releasing a perceptually-uniform color map.

ASJC Scopus subject areas

  • Geochemistry and Petrology
  • Geophysics


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