1887
Volume 24, Issue 1
  • ISSN: 1569-4445
  • E-ISSN: 1873-0604

Abstract

Abstract

Solving the elastic or acoustic wave equation is essential for seismic imaging and inversion techniques. Although conventional methods, like finite difference or finite element schemes, are widely used, they suffer from low computational efficiency, especially in large‐scale applications. To overcome this limitation, we propose a novel deep learning‐based framework using Fourier neural operators (FNOs), which learn mappings from geological parameters to wavefield solutions. By integrating finite difference simulations with stochastic medium modelling, we generated training datasets encompassing diverse geological conditions. The neural operator was iteratively optimized through targeted training trials to enhance its predictive capability. The resulting operator achieves high accuracy ( error: 0.05–0.30) while preserving numerical fidelity comparable to traditional methods. More notably, the operator offers significant speedups, 170‐fold for acoustic and 260‐fold for elastic wave equations. Validated through comprehensive experiments, this operator serves as an efficient and reliable input for downstream seismic processing workflows, enabling end‐to‐end acceleration in seismic waveform inversion and imaging systems.

Loading

Article metrics loading...

/content/journals/10.1002/nsg.70033
2026-01-24
2026-02-16
Loading full text...

Full text loading...

References

  1. Alkhalifah, T., Song, C., Waheed, U.B. & Hao, Q. (2021) Wavefield solutions from machine learned functions constrained by the Helmholtz equation. Artificial Intelligence in Geosciences, 2, 11–19.
    [Google Scholar]
  2. Belkin, M., Hsu, D., Ma, S.Y. & Mandal, S. (2019) Reconciling modern machine‐learning practice and the classical bias‐variance trade‐off. PNAS, 116(32), 15849–15854..
    [Google Scholar]
  3. bin Waheed, U.B., Haghighat, E., Alkhalifah, T., Song, C. & Hao, Q. (2021) PINNeik: eikonal solution using physics‐informed neural networks. Computers and Geosciences, 155, 104833.
    [Google Scholar]
  4. Chang, H. & Zhang, D. (2019) Identification of physical processes via combined data‐driven and data‐assimilation methods. Journal of Computational Physics, 393, 337–350.
    [Google Scholar]
  5. Chen, Y., Lu, L., Karniadakis, G.E. & Dal Negro, L. (2020) Physics‐informed neural networks for inverse problems in nano‐optics and metamaterials. Optics Express, 28, 11618.
    [Google Scholar]
  6. Dal Santo, N., Deparis, S. & Pegolotti, L. (2020) Data driven approximation of parametrized PDEs by reduced basis and neural networks. Journal of Computational Physics, 416, 109550.
    [Google Scholar]
  7. Erichson, N.B., Muehlebach, M. & Mahoney, M.W. (2019) Physics‐informed autoencoders for lyapunov‐stable fluid flow prediction. arXiv preprint. https://doi.org/10.48550/arXiv.1905.10866
    [Google Scholar]
  8. Esmaeilzadeh, S., Azizzadenesheli, K., Kashinath, K., Mustafa, M., Tchelepi, H.A., Marcus, P., et al. (2020) Meshfree‐FlowNet: a physics‐constrained deep continuous space‐time super‐resolution framework. In: SC20: International conference for high performance computing, networking, storage and analysis. New York: ACM Publication, pp. 1–15.
  9. Gupta, G., Xiao, X. & Bogdan, P. (2021) Multiwavelet‐based operator learning for differential equations. Advances in Neural Information Processing Systems, 34, 24048–24062.
    [Google Scholar]
  10. Huang, X. & Alkhalifah, T. (2021) PINNup: robust neural network wavefield solutions using frequency upscaling and neuron splitting. Journal of Geophysical Research: Solid Earth, 127, e2021JB023703.
    [Google Scholar]
  11. Ikelle, L.T., Yung, S.K. & Daube, F. (1993) 2‐D random media with ellipsoidal autocorrelation functions. Geophysics, 58(9), 1359–1372.
    [Google Scholar]
  12. Kingma, D.P. & Ba, J.L. (2014) Adam: a method for stochastic optimization. arXiv preprint. https://doi.org/10.48550/arXiv.1412.6980
    [Google Scholar]
  13. Kochkov, D., Smith, J.A., Alieva, A., Wang, Q., Brenner, M.P. & Hoyer, S. (2021) Machine learning‐accelerated computational fluid dynamics. Proceedings of the National Academy of Sciences, 118, e2101784118.
    [Google Scholar]
  14. Kovachki, N., Lanthaler, S. & Mishra, S. (2021) On universal approximation and error bounds for Fourier neural operators. Journal of Machine Learning Research, 22, 1–76.
    [Google Scholar]
  15. Li, Z.Y., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A. et al. (2020) Fourier neural operator for parametric partial differential equations. arXiv preprint. https://doi.org/10.48550/arXiv.2010.08895
    [Google Scholar]
  16. Lu, L., Jin, P. & Karniadakis, G.E. (2019) DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. arXiv preprint.https://doi.org/10.48550/arXiv.1910.03193
    [Google Scholar]
  17. Moseley, B., Markham, A. & Nissen‐Meyer, T. (2020) Solving the wave equation with physics‐informed deep learning. arXiv preprint. https://doi.org/10.48550/arXiv.2006.11894
    [Google Scholar]
  18. Ong, Y.Z., Shen, Z. & Yang, H. (2022) IAE‐Net: integral autoencoders for discretization‐invariant learning. arXiv preprint. https://doi.org/10.48550/arXiv.2203.05142
    [Google Scholar]
  19. Pratt, R.G. (1990) Inverse theory applied to multi‐source cross‐hole tomography. Part 2: elastic wave‐equation method. Geophysical Prospecting, 38, 311–329.
    [Google Scholar]
  20. Rahman, M.A., Ross, Z.E. & Azizzadenesheli, K. (2022) U‐No: U‐shaped neural operators. arXiv preprint.https://doi.org/10.48550/arXIV.2204.11127
    [Google Scholar]
  21. Raissi, M., Perdikaris, P. & Karniadakis, G.E. (2017) Physics informed deep learning (part I): data‐driven solutions of nonlinear partial differential equations. arXiv preprint.https://doi.org/10.48550/arXiv.1711.10561
    [Google Scholar]
  22. Raissi, M., Perdikaris, P. & Karniadakis, G.E. (2019) Physics‐informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707.
    [Google Scholar]
  23. Ronneberger, O., Fischer, P. & Brox, T. (2015) U‐Net: convolutional networks for biomedical image segmentation. arXiv preprint.https://doi.org/10.48550/arXiv.1505.04597
    [Google Scholar]
  24. Sirignano, J. & Spiliopoulos, K. (2018) DGM: a deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 375, 1339–1364.
    [Google Scholar]
  25. Smith, J.D., Azizzadenesheli, K. & Ross, Z.E. (2020) Eikonet: solving the eikonal equation with deep neural networks. IEEE Transactions on Geoscience and Remote Sensing, 59, 10685–10696.
    [Google Scholar]
  26. Song, C., Alkhalifah, T. & Waheed, U.B. (2021) Solving the frequency domain acoustic VTI wave equation using physics‐informed neural networks. Geophysical Journal International, 225, 846–859.
    [Google Scholar]
  27. Sun, B.B. & Alkhalifah, T. (2020) ML‐descent: an optimization algorithm for full‐waveform inversion using machine learning. Geophysics, 85(6), R477–R492.
    [Google Scholar]
  28. Sun, J., Niu, Z., Innanen, K.A., Li, J. & Trad, D.O. (2020) A theory‐guided deep‐learning formulation and optimization of seismic waveform inversion. Geophysics, 85(2), R87–R99.
    [Google Scholar]
  29. Tripura, T. & Chakraborty, S. (2022) Wavelet neural operator: a neural operator for parametric partial differential equations. arXiv preprint. https://doi.org/10.48550/arXiv.2205.02191.
  30. Virieux, J. (1986) P‐SV wave propagation in heterogeneous media: velocity stress finite‐difference method. Geophysics, 51, 889–901.
    [Google Scholar]
  31. Wang, L.M., Luo, Y.H. & Xu, X.Y. (2012) Numerical investigation of Rayleigh‐wave propagation on topography surface. Journal of Applied Geophysics, 86, 88–97.
    [Google Scholar]
  32. Wang, Q.Q., Song, P., Hua, Q.F., Liu, B., Li, G., Wang, S. et al. (2023) Full waveform inversion based on Adam algorithm with optimized parameters. Chinese Journal of Geophysics, 66(11), 4654–4663.
    [Google Scholar]
  33. Wang, S.F., Wang, H.W. & Perdikaris, P. (2021) Learning the solution operator of parametric partial differential equations with physics‐informed DeepONets. Science Advances, 7(40), eabi8605.
    [Google Scholar]
  34. Wei, W. & Fu, L.Y. (2022) Small‐data‐driven fast seismic simulations for complex media using physics‐informed Fourier neural operators. Geophysics, 87(6), T435–T446.
    [Google Scholar]
  35. Winovich, N., Ramani, K. & Lin, G. (2019) ConvPDE‐UQ: convolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains. Journal of Computational Physics, 394, 263–279.
    [Google Scholar]
  36. Xiao, C., Deng, Y. & Wang, G. (2021) Deep‐learning‐based adjoint state method: methodology and preliminary application to inverse modeling. Water Resources Research, 57, 2020WR027400.
    [Google Scholar]
  37. Yu, B. (2018) The deep Ritz method: a deep learning‐based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 6, 1–12.
    [Google Scholar]
  38. Zhang, T., Trad, D. & Innanen, K. (2023) Learning to solve the elastic wave equation with Fourier neural operators. Geophysics, 88(3), T101–T119.
    [Google Scholar]
  39. Zhu, W., Xu, K., Darve, E. & Beroza, G.C. (2021) A general approach to seismic inversion with automatic differentiation. Computers and Geosciences, 151, 104751.
    [Google Scholar]
/content/journals/10.1002/nsg.70033
Loading
/content/journals/10.1002/nsg.70033
Loading

Data & Media loading...

  • Article Type: Research Article
Keyword(s): modelling; seismic; wavefield

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error