Wavefield solutions from machine learned functions constrained by the Helmholtz equation

Tariq Alkhalifah*, Chao Song, Umair bin Waheed, Qi Hao

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

39 Scopus citations


Solving the wave equation is one of the most (if not the most) fundamental problems we face as we try to illuminate the Earth using recorded seismic data. The Helmholtz equation provides wavefield solutions that are dimensionally reduced, per frequency, compared to the time domain, which is useful for many applications, like full waveform inversion. However, our ability to attain such wavefield solutions depends often on the size of the model and the complexity of the wave equation. Thus, we use here a recently introduced framework based on neural networks to predict functional solutions through setting the underlying physical equation as a loss function to optimize the neural network (NN) parameters. For an input given by a location in the model space, the network learns to predict the wavefield value at that location, and its partial derivatives using a concept referred to as automatic differentiation, to fit, in our case, a form of the Helmholtz equation. We specifically seek the solution of the scattered wavefield considering a simple homogeneous background model that allows for analytical solutions of the background wavefield. Providing the NN with a reasonable number of random points from the model space will ultimately train a fully connected deep NN to predict the scattered wavefield function. The size of the network depends mainly on the complexity of the desired wavefield, with such complexity increasing with increasing frequency and increasing model complexity. However, smaller networks can provide smoother wavefields that might be useful for inversion applications. Preliminary tests on a two-box-shaped scatterer model with a source in the middle, as well as, the Marmousi model with a source at the surface demonstrate the potential of the NN for this application. Additional tests on a 3D model demonstrate the potential versatility of the approach.

Original languageEnglish (US)
Pages (from-to)11-19
Number of pages9
JournalArtificial Intelligence in Geosciences
StatePublished - Dec 2021

Bibliographical note

Publisher Copyright:
© 2021 The Authors


  • Deep learning
  • Helmholtz equation
  • Modeling
  • Neural networks
  • Wavefields

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computers in Earth Sciences
  • Earth and Planetary Sciences (miscellaneous)
  • Artificial Intelligence


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