https://doi.org/10.26089/NumMet.v27r437
Deep learning in joint 3D inversion of gravitational and magnetic data: comparison of loss functions
Keywords:
Abstract
The paper proposes a neural network approach to the joint three-dimensional inversion of gravitational and magnetic data aimed at the recovery of a single ore body. The method is based on a convolutional U-Net architecture trained on a large synthetic dataset generated by stochastic combinations of rectangular prisms simulating typical geological structures. Various loss functions are considered and compared: mean squared error (MSE), mean absolute error (MAE), Huber loss, and a loss with a forward-problem physics residual. It is shown that, in the conducted synthetic experiments, this loss provides the lowest error among the considered variants and improves the physical consistency of the recovered density and magnetization models, while data augmentation by rotations improves the reconstruction quality. The results of neural network recovery are also compared with the classical solution of the inverse problem within the framework of discrete potential theory. In the considered test example, the neural network approach achieved an error of 4.23% for the loss with a physics residual with a processing time of 0.2 s on the fixed 21³ grid.
Published
Issue
Section
References
- Y. Hu, X. Wei, X. Wu, et al., “A deep learning-enhanced framework for multiphysics joint inversion,” Geophysics 88 (1), K13–K26 (2023).
doi 10.1190/geo2021-0589.1 - S. Vatankhah, X. Huang, R. A. Renaut, et al., “Joint Gramian inversion of gravity, magnetic and gravity gradient tensor data for the modelling of Mesoproterozoic iron oxide deposits, southeast Missouri, USA,” Geophys. J. Int. 235 (2), 1064–1085 (2023).
doi 10.1093/gji/ggad290 - S. Liu, X. Wan, S. Jin, et al., “Fast 3D joint inversion of gravity and magnetic data based on cross-gradient constraint,” Geodesy and Geodynamics 14 (4), 331–346 (2023).
doi 10.1016/j.geog.2022.12.003 - L. A. Gallardo and M. A. Meju, “Joint two-dimensional DC resistivity and seismic travel time inversion with cross-gradients constraints,” J. Geophys. Res.: Solid Earth 109 (B3) (2004).
doi 10.1029/2003JB002716 - N. Wei, D. Yang, Z. Wang, and Y. Lu, “Joint 3D Inversion of Gravity and Magnetic Data Using Deep Learning Neural Networks,” in Proceedings of the Second International Meeting for Applied Geoscience & Energy, Houston, Texas, USA, August 28 – September 1, 2022(ASME, 2022), pp. 1457–1461.
doi 10.1190/image2022-3751223.1 - Z. Zhou, J. Wang, X. Meng, and Y. Fang, “3-D Joint Inversion of Gravity and Magnetic Data Based on a Deep Learning Network With Automatic Recognition of Structural Similarity,” IEEE Trans. Geosci. Remote Sens. 61, 1–16 (2023).
doi 10.1109/TGRS.2023.3330988 - Y. Li, W. Lu, Z. Jia, and C. Song, “Two-Stage Deep Learning for 3D Joint Inversion of Gravity and Magnetic,” in Proceedings of the Third International Meeting for Applied Geoscience & Energy, Houston, Texas, USA, August 27 – September 1, 2023(ASME, 2023), pp. 1030–1034.
doi 10.1190/image2023-3900861.1 - J. Jiao, S. Dong, S. Zhou, et al., “3-D Gravity and Magnetic Joint Inversion Based on Deep Learning Combined with Measurement Data Constraint,” IEEE Trans. Geosci. Remote Sens. 62, 1–14 (2024).
doi 10.1109/TGRS.2023.3339303 - Z. Bai, Y. Wang, C. Wang, et al., “Joint Gravity and Magnetic Inversion Using CNNs’ Deep Learning,” Remote Sensing 16 (7), Article Number 1115 (2024).
doi 10.3390/rs16071115 - Y. Wang, D. V. Churbanov, R. L. Argun, et al., “A structural two-level neural network approach to joint inversion of gravitational and magnetic fields,” Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie) 26 (3), 322–339 (2025).
doi 10.26089/NumMet.v26r322 - G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Mathematics of Control, Signals, and Systems 2 (4), 303–314 (1989).
doi 10.1007/BF02551274 - O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional Networks for Biomedical Image Segmentation,” in Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015, Lecture Notes in Computer Science, Vol. 9351(Springer, Cham, 2015), pp. 234–241.
doi 10.1007/978-3-319-24574-4_28 - M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations,” J. Comput. Phys. 378, 686–707 (2019).
doi 10.1016/j.jcp.2018.10.045 - NVIDIA Deep Learning Performance.
https://docs.nvidia.com/deeplearning/performance/mixed-precision-training/index.html Cited September 19, 2026. - I. Loshchilov and F. Hutter, “Decoupled Weight Decay Regularization,” International Conference on Learning Representations (ICLR), 2019.
https://openreview.net/forum?id=Bkg6RiCqY7 Cited September 19, 2026. - I. E. Stepanova, I. I. Kolotov, A. N. Levashov, and A. V. Gorbachev, “On Finding Stable Approximate Solutions of Linear Inverse Problems of Gravimetry in the Discrete Potential Theory: A Local Version,” Izvestiya, Physics of the Solid Earth 61 (1), 126–141 (2025).
doi 10.1134/S1069351325700089
License
Copyright (c) 2026 А. В. Горбачев, А. Н. Левашов, Д. В. Чурбанов, И. Э. Степанова
This work is licensed under a Creative Commons Attribution 4.0 International License.
