Application of neural networks to inverse elastic scattering problems with near-field measurements

Yao Sun; Lijuan He; Bo Chen; Yao Sun; Lijuan He; Bo Chen

doi:10.3934/era.2023355

Electronic Research Archive

2023, Volume 31, Issue 11: 7000-7020. doi: 10.3934/era.2023355

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Research article

Application of neural networks to inverse elastic scattering problems with near-field measurements

College of Science, Civil Aviation University of China, Tianjin, China

Received: 26 August 2023 Revised: 12 October 2023 Accepted: 24 October 2023 Published: 01 November 2023

This paper is concerned with the application of a machine learning approach to inverse elastic scattering problems via neural networks. In the forward problem, the displacements are approximated by linear combinations of the fundamental tensors of the Cauchy-Navier equations of elasticity, which are expressed in terms of sources placed inside the elastic solid. From the near-field measurement data, a two-layer neural network method consisting of a gated recurrent unit to gate recurrent unit has been used to reconstruct the shape of an unknown elastic body. Moreover, the convergence of the method is proved. Finally, the feasibility and effectiveness of the presented method are examined through numerical examples.
- partial differential equations,
- inverse scattering problem,
- linear elasticity
Citation: Yao Sun, Lijuan He, Bo Chen. Application of neural networks to inverse elastic scattering problems with near-field measurements[J]. Electronic Research Archive, 2023, 31(11): 7000-7020. doi: 10.3934/era.2023355

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Abstract

This paper is concerned with the application of a machine learning approach to inverse elastic scattering problems via neural networks. In the forward problem, the displacements are approximated by linear combinations of the fundamental tensors of the Cauchy-Navier equations of elasticity, which are expressed in terms of sources placed inside the elastic solid. From the near-field measurement data, a two-layer neural network method consisting of a gated recurrent unit to gate recurrent unit has been used to reconstruct the shape of an unknown elastic body. Moreover, the convergence of the method is proved. Finally, the feasibility and effectiveness of the presented method are examined through numerical examples.

References

[1]	G. Bao, T. Yin, Recent progress on the study of direct and inverse elastic scattering problems (in Chinese), Sci. Sin. Math., 47 (2017), 1103–1118. https://doi.org/10.1360/N012016-00198 doi: 10.1360/N012016-00198
[2]	D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer Nature, Berlin, 2019. https://doi.org/10.1007/978-3-030-30351-8
[3]	F. Cakoni, D. Colton, A Qualitative Approach to Inverse Scattering Theory, Springer, US, 2014. https://doi.org/10.1007/978-1-4614-8827-9
[4]	F. Cakoni, D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer, Vienna, 2006. https://doi.org/10.1007/3-540-31230-7
[5]	H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee, A. Wahab, Mathematical Methods in Elasticity Imaging, Princeton University Press, New Jersey, 2015.
[6]	J. H. Bramble, J. E. Pasciak, A note on the existence and uniqueness of solutions of frequency domain elastic wave problems: a priori estimates in $H^1$, J. Math. Anal. Appl., 345 (2008), 396–404. https://doi.org/10.1016/j.jmaa.2008.04.028 doi: 10.1016/j.jmaa.2008.04.028
[7]	G. Bao, G. Hu, J. Sun, T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263–301. https://doi.org/10.1016/j.matpur.2018.01.007 doi: 10.1016/j.matpur.2018.01.007
[8]	H. Liu, M. Petrini, L. Rondi, J. Xiao, Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements, J. Differ. Equations, 262 (2017), 1631–1670. https://doi.org/10.1016/j.jde.2016.10.021 doi: 10.1016/j.jde.2016.10.021
[9]	J. Li, H. Liu, Y. Wang, Recovering an electromagnetic obstacle by a few phaseless backscattering measurements, Inverse Probl., 333 (2017), 035011. https://doi.org/10.1088/1361-6420/aa5bf3 doi: 10.1088/1361-6420/aa5bf3
[10]	J. Li, H. Liu, Q. Wang, Enhanced multilevel linear sampling methods for inverse scattering problems, J. Comput. Phys., 257 (2014), 554–571. https://doi.org/10.1016/j.jcp.2013.09.048 doi: 10.1016/j.jcp.2013.09.048
[11]	T. Arens, Linear sampling methods for 2D inverse elastic wave scattering, Inverse Probl., 17 (2001), 1445–1464. https://doi.org/10.1088/0266-5611/17/5/314 doi: 10.1088/0266-5611/17/5/314
[12]	A. Charalambopoulos, A. Kirsch, K. Anagnostopoulos, D. Gintides, K. Kiriaki, The factorization method in inverse elastic scattering from penetrable bodies, Inverse Probl., 23 (2007), 27–51. https://doi.org/10.1088/0266-5611/23/1/002 doi: 10.1088/0266-5611/23/1/002
[13]	G. Hu, A. Kirsch, M. Sini, Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Probl., 29 (2013), 015009. https://doi.org/10.1088/0266-5611/29/1/015009 doi: 10.1088/0266-5611/29/1/015009
[14]	Z. Chen, G. Huang, Reverse time migration for extended obstacles: elastic waves (in Chinese), Sci. Sin. Math., 45 (2015), 1103–1114. https://doi.org/10.1360/N012014-00097 doi: 10.1360/N012014-00097
[15]	P. Li, Y. Wang, Z. Wang, Y. Zhao, Inverse obstacle scattering for elastic waves, Inverse Probl., 32 (2016), 115018. https://doi.org/10.1088/0266-5611/32/11/115018 doi: 10.1088/0266-5611/32/11/115018
[16]	J. Ant$\acute{o}$nio, A. Tadeu, L. Godinho, A three-dimensional acoustics model using the method of fundamental solutions, Eng. Anal. Boundary Elem., 32 (2008), 525–531. https://doi.org/10.1016/j.enganabound.2007.10.008 doi: 10.1016/j.enganabound.2007.10.008
[17]	A. Karageorghis, B. T. Johansson, D. Lesnic, The method of fundamental solutions for the identification of a sound-soft obstacle in inverse acoustic scattering, Appl. Numer. Math., 62 (2012), 1767–1780. https://doi.org/10.1016/j.apnum.2012.05.011 doi: 10.1016/j.apnum.2012.05.011
[18]	Y. Sun, X. Lu, B. Chen, The method of fundamental solutions for the high frequency acoustic-elastic problem and its relationship to a pure acoustic problem, Eng. Anal. Boundary Elem., 156 (2023), 299–310. https://doi.org/10.1016/j.enganabound.2023.08.010 doi: 10.1016/j.enganabound.2023.08.010
[19]	A. Karageorghis, D. Lesnic, L. Marin, The MFS for the identification of a sound-soft interior acoustic scatterer, Eng. Anal. Boundary Elem., 83 (2017), 107–112. https://doi.org/10.1016/j.enganabound.2017.07.021 doi: 10.1016/j.enganabound.2017.07.021
[20]	A. Karageorghis, D. Lesnic, L. Marin, The method of fundamental solutions for the identification of a scatterer with impedance boundary condition in interior inverse acoustic scattering, Eng. Anal. Boundary Elem., 92 (2018), 218–224. https://doi.org/10.1016/j.enganabound.2017.07.005 doi: 10.1016/j.enganabound.2017.07.005
[21]	D. Colton, R. Kress, Using fundamental solutions in inverse scattering, Inverse Probl., 22 (2006), R49. https://doi.org/10.1088/0266-5611/22/3/R01 doi: 10.1088/0266-5611/22/3/R01
[22]	C. J. S. Alves, N. F. M. Martins, S. S. Valtchev, Extending the method of fundamental solutions to non-homogeneous elastic wave problems, Appl. Numer. Math., 115 (2017), 299–313. https://doi.org/10.1016/j.apnum.2016.06.002 doi: 10.1016/j.apnum.2016.06.002
[23]	W. Yin, W. Yang, H. Liu, A neural network scheme for recovering scattering obstacles with limited phaseless far-field data, J. Comput. Phys., 417 (2020), 109594. https://doi.org/10.1016/j.jcp.2020.109594 doi: 10.1016/j.jcp.2020.109594
[24]	Z. Wei, X. Chen, Deep-learning schemes for full-wave nonliner inverse scattering problems, IEEE Trans. Geosci. Remote Sens., 57 (2019), 1849–1860. https://doi.org/10.1109/TGRS.2018.2869221 doi: 10.1109/TGRS.2018.2869221
[25]	E. Chung, J. G. Park, Sentence-chain based Seq2seq model for corpus expansion, ETRI J., 39 (2017), 455–466. https://doi.org/10.4218/etrij.17.0116.0074 doi: 10.4218/etrij.17.0116.0074
[26]	J. Torres, C. Vaca, L. Terán, C. L. Abad, Seq2Seq models for recommending short text conversations, Expert Syst. Appl., 150 (2020), 113270. https://doi.org/10.1016/j.eswa.2020.113270 doi: 10.1016/j.eswa.2020.113270
[27]	Y. Khoo, L. Ying, SwitchNet: a neural network model for forward and inverse scattering problems, SIAM J. Sci. Comput., 41 (2019), A3182–A3201. https://doi.org/10.1137/18M1222399 doi: 10.1137/18M1222399
[28]	G. Fairweather, A. Karageorghis, P. A. Martin, The method of fundamental solutions for scattering and radiation problems, Eng. Anal. Boundary Elem., 27 (2003), 759–769. https://doi.org/10.1016/S0955-7997(03)00017-1 doi: 10.1016/S0955-7997(03)00017-1
[29]	Y. Sun, F. Ma, A meshless method for the Cauchy problem in linear elastodynamics, Appl. Anal., 93 (2014), 2647–2667. https://doi.org/10.1080/00036811.2014.882913 doi: 10.1080/00036811.2014.882913
[30]	P. Zhang, P. Meng, W. Yin, H. Liu, A neural network method for time-dependent inverse source problem with limited-aperture data, J. Comput. Appl. Math., 421 (2023), 114842. https://doi.org/10.1016/j.cam.2022.114842 doi: 10.1016/j.cam.2022.114842
[31]	D. Xu, Z. Li, W. Wu, X. Ding, D. Qu, Convergence of gradient method descent algorithm for recurrent neuron, Int. Symp. Neural Network, 4493 (2007), 117–122. https://doi.org/10.1007/978-3-540-72395-0_16 doi: 10.1007/978-3-540-72395-0_16
[32]	D. Xu, Z. Li, W. Wu, Convergence of gradient method for fully recurrent neural network, Soft Comput., 14 (2010), 245–250. https://doi.org/10.1007/s00500-009-0398-0 doi: 10.1007/s00500-009-0398-0

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