ON THE SOLUTION OF BOUNDARY-VALUE PROBLEMS OF PHYSICAL GEODESY IN THE FORM OF DEEP NEURAL NETWORKS
1 Moscow State University of Geodesy and Cartography
2 Schmidt Institute of Physics of the Earth, Russian Academy of Sciences
Journal: Geophysical research
Tome: 25
Number: 2
Year: 2024
Pages: 5-19
UDK: 550.831.015: 550.831.23: 519.654
DOI: 10.21455/gr2024.2-1
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Neyman
L.S Yu.M. ON THE SOLUTION OF BOUNDARY-VALUE PROBLEMS OF PHYSICAL GEODESY IN THE FORM OF DEEP NEURAL NETWORKS
// . 2024. Т. 25. № 2. С. 5-19. DOI: 10.21455/gr2024.2-1
@article{Neyman
L.SON2024,
author = "Neyman
L.S, Yu. M.",
title = "ON THE SOLUTION OF BOUNDARY-VALUE PROBLEMS OF PHYSICAL GEODESY IN THE FORM OF DEEP NEURAL NETWORKS
",
journal = "Geophysical research",
year = 2024,
volume = "25",
number = "2",
pages = "5-19",
doi = "10.21455/gr2024.2-1",
language = "English"
}
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Keywords: Earth's gravitational field, modeling, boundary value problem, neural network, deep learning.
Аnnotation: The development of a new approach to the local approximation of the anomalous gravitational field of the Earth is discussed. It is based on the integration of modern global geopotential models used to represent the low-frequency part of gravitational anomalies with the models of the high-frequency component described by the solution of the Laplace differential equation in the local area. To obtain a solution to the Laplace equation, it is proposed to use pseudoboundary conditions based on the results of geodetic measurements of the geopotential transforms in a local area, including airborne gravimetry data. Additionally, it is recommended to take into ac-count the effect of smoothing the geopotential with height. Taking into account this effect comes down to the fact that, with modern requirements for the accuracy of determining gravitational anomalies, it allows, already at a relatively small distance from the Earth's surface, depending on the dimension of the reference low-frequency geopotential model, to consider its high-frequency component to be a sufficiently small value, which can practically be taken equal to zero. In fact, this is the result of the practical application of such a property of the geopotential as regularity at infinity. The possibility of such a replacement is confirmed by the results of a computational experiment using the global geopotential model XGM2019e in the form of spherical harmonics up to the 5540th degree. The main feature of the proposed approach to modeling the high-frequency component is the search for a solution to the Laplace differential equation in the form of a deep artificial neural network that ensures the fulfillment of a certain optimization condition. In this case, the auxiliary problem of calculating partial derivatives can be effectively solved using the currently known technique of automatic computer differentiation. The modification of the artificial neural network parameters with sufficient reliability is ensured by the algorithm for estimating stochastic moments (Adam). In the final part of the article, the prospects for continuing research into the study of the Earth's gravitational field using mathematical modeling based on the described methodology are noted.
Bibliography: Antoniou A., Lu W.-S., Practical Optimization: Algorithms and Engineering Applications, New York, Springer, 2007, 670 p. DOI: 10.1007/978-1-0716-0843-2
Cybenko G.V., Approximation by Superpositions of a Sigmoidal function, Mathematics of Control, Signals, and Systems, 1989, vol. 2, pp. 303-314.
Funahashi K., On the Approximate Realization of Continuous Mappings by Neural Networks, Neural Networks, 1989, vol. 2, pp. 183-192.
Grushinskiy N.P., Teoriya figury Zemli (Theory of the figure of the Earth), Moscow, Nauka, 1976, 512 p. [In Russian].
Goodfellow I., Bengio Y., Courville A., Deep Learning, Cambridge, MA, MIT Press, 2016, 800 p.
Hastings D., Dunbar P.K., Global Land One-kilometer Base Elevation (GLOBE) v.1, Boulder, Colorado, National Geophysical Data Center, NOAA, 1999. 147 p. DOI: 10.7289/V52R3PMS
He K., Zhang X., Ren S., Sun J., Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification, in Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2015, pp. 1026-1034. DOI: 10.1109/ICCV.2015
Hofmann-Wellenhof B., Moritz H., Physical Geodesy, Vienna, Springer Vienna, 2005, 403 p.
Hornik K., Stinchcombe M., White H., Multilayer feedforward networks are universal approximators, Neural Network, 1989, vol. 2, pp. 359-366.
Kallan R., Osnovnyye kontseptsii neyronnykh setey (Basic concepts of neural networks), Moscow, Izdatel'skii dom “Vil'yams”, 2001, 287 p. [In Russian].
Kingma D.P., Ba L.J., ADAM: A Method For Stochastic Optimization. arXiv, 2014. URL: http://arxiv.org/ abs/1412.6980v9. Accessed 23.08.2022.
Koch K.R., Pope A.J., Uniqueness and existence for the geodetic boundary value problem using the known sur-face of the Earth, Bull. Geod., 1972, vol. 106, pp. 467-476.
Leshno M., Lin V., Pinkus A., Schocken S., Multilayer feedforward networks with nonpolynomial activation function can approximate any function, Neural Networks, 1993, vol. 6, pp. 861-867.
Li H., Chen S., Li Y., Zhang B., Zhao M., Han J., Stable downward continuation of the gravity potential field implemented using deep learning, Frontiers in Earth Science, 2023, vol. 10, 13 p.
Nepoklonov V.B., Computer models of the Earth's anomalous gravitational field, Izvestiya vuzov. Geodeziya i aerofotos"yemka (News of higher educational institutions. Geodesy and aerial photography), 1998, no. 6, pp. 104-111. [In Russian].
Neyman Y.M., Sugaipova L.S., Solving the Laplace differential equation in the form of a deep neural network as a unified algorithm for the approximate solution of problems of physical geodesy in a local area, Izvestiya vuzov. Geodeziya i aerofotos"yemka (News of higher educational institutions. Geodesy and aeri-al photography), 2023, vol. 67, no. 1, pp. 18-25. [In Russian]. DOI: 30533/GiA-2023-002
Nikolenko S., Kadurin A., Arkhangel’skaya E., Glubokoe obuchenie (Deep learning), Saint Petersburg, Piter, 2018, 480 p. [In Russian].
Raissi M., Perdikaris P., Karniadakis G.E., Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Com-putational Physics, 2019, vol. 378, pp. 686-707. DOI: 10.1016/j.jcp.2018.10.045
Tikhonov A.N., Samarskiy A.A., Uravneniya matematicheskoy fiziki (Equations of mathematical physics), Moscow, Nauka, 1977, 736 р. [In Russian].
Voronina V.V., Mikheyev A.V., Yarushkina N.G., Svyatov K.V., Teoriya i praktika mashinnogo obucheniya (Theory and practice of machine learning), Ulyanovsk, UlGTU, 2017, 290 p. [In Russian].
Zingerle P., Pail R., Gruber T., Oikonomidou X., The combined global gravity field model XGM2019e, J. Geod, 2020, vol. 94, 12 p. DOI: 10.1007/s00190-020-01398-0
Cybenko G.V., Approximation by Superpositions of a Sigmoidal function, Mathematics of Control, Signals, and Systems, 1989, vol. 2, pp. 303-314.
Funahashi K., On the Approximate Realization of Continuous Mappings by Neural Networks, Neural Networks, 1989, vol. 2, pp. 183-192.
Grushinskiy N.P., Teoriya figury Zemli (Theory of the figure of the Earth), Moscow, Nauka, 1976, 512 p. [In Russian].
Goodfellow I., Bengio Y., Courville A., Deep Learning, Cambridge, MA, MIT Press, 2016, 800 p.
Hastings D., Dunbar P.K., Global Land One-kilometer Base Elevation (GLOBE) v.1, Boulder, Colorado, National Geophysical Data Center, NOAA, 1999. 147 p. DOI: 10.7289/V52R3PMS
He K., Zhang X., Ren S., Sun J., Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification, in Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2015, pp. 1026-1034. DOI: 10.1109/ICCV.2015
Hofmann-Wellenhof B., Moritz H., Physical Geodesy, Vienna, Springer Vienna, 2005, 403 p.
Hornik K., Stinchcombe M., White H., Multilayer feedforward networks are universal approximators, Neural Network, 1989, vol. 2, pp. 359-366.
Kallan R., Osnovnyye kontseptsii neyronnykh setey (Basic concepts of neural networks), Moscow, Izdatel'skii dom “Vil'yams”, 2001, 287 p. [In Russian].
Kingma D.P., Ba L.J., ADAM: A Method For Stochastic Optimization. arXiv, 2014. URL: http://arxiv.org/ abs/1412.6980v9. Accessed 23.08.2022.
Koch K.R., Pope A.J., Uniqueness and existence for the geodetic boundary value problem using the known sur-face of the Earth, Bull. Geod., 1972, vol. 106, pp. 467-476.
Leshno M., Lin V., Pinkus A., Schocken S., Multilayer feedforward networks with nonpolynomial activation function can approximate any function, Neural Networks, 1993, vol. 6, pp. 861-867.
Li H., Chen S., Li Y., Zhang B., Zhao M., Han J., Stable downward continuation of the gravity potential field implemented using deep learning, Frontiers in Earth Science, 2023, vol. 10, 13 p.
Nepoklonov V.B., Computer models of the Earth's anomalous gravitational field, Izvestiya vuzov. Geodeziya i aerofotos"yemka (News of higher educational institutions. Geodesy and aerial photography), 1998, no. 6, pp. 104-111. [In Russian].
Neyman Y.M., Sugaipova L.S., Solving the Laplace differential equation in the form of a deep neural network as a unified algorithm for the approximate solution of problems of physical geodesy in a local area, Izvestiya vuzov. Geodeziya i aerofotos"yemka (News of higher educational institutions. Geodesy and aeri-al photography), 2023, vol. 67, no. 1, pp. 18-25. [In Russian]. DOI: 30533/GiA-2023-002
Nikolenko S., Kadurin A., Arkhangel’skaya E., Glubokoe obuchenie (Deep learning), Saint Petersburg, Piter, 2018, 480 p. [In Russian].
Raissi M., Perdikaris P., Karniadakis G.E., Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Com-putational Physics, 2019, vol. 378, pp. 686-707. DOI: 10.1016/j.jcp.2018.10.045
Tikhonov A.N., Samarskiy A.A., Uravneniya matematicheskoy fiziki (Equations of mathematical physics), Moscow, Nauka, 1977, 736 р. [In Russian].
Voronina V.V., Mikheyev A.V., Yarushkina N.G., Svyatov K.V., Teoriya i praktika mashinnogo obucheniya (Theory and practice of machine learning), Ulyanovsk, UlGTU, 2017, 290 p. [In Russian].
Zingerle P., Pail R., Gruber T., Oikonomidou X., The combined global gravity field model XGM2019e, J. Geod, 2020, vol. 94, 12 p. DOI: 10.1007/s00190-020-01398-0