# American Institute of Mathematical Sciences

June  2020, 28(2): 1123-1142. doi: 10.3934/era.2020062

## A neural network method for the inverse scattering problem of impenetrable cavities

 1 Experiment Centre of Mathematics, Changchun University of Science and Technology, Changchun 130022, China 2 College of Science, Changchun University of Science and Technology, Changchun 130022, China 3 School of Computer Science and Technology, Changchun University, of Science and Technology, Changchun 130022, China

* Corresponding author: Weishi Yin

Received  May 2020 Revised  May 2020 Published  June 2020

Fund Project: The research of first author is supported by National Natural Science Foundation of China grant 11671107

This paper proposes a near-field shape neural network (NSNN) to determine the shape of a sound-soft cavity based on a single source and several measurements placed on a curve inside the cavity. The NSNN employs the near-field measurements as input, and the output is the shape parameters of the cavity. The self-attention mechanism is employed to obtain the feature information of the near-field data, as well as the correlations among them. The weights and biases of the NSNN are updated through the gradient descent algorithm, which minimizes the error of the reconstructed shape of the cavity. We prove that the loss function sequence related to the weights is a monotonically bounded non-negative sequence, which indicates the convergence of the NSNN. Numerical experiments show that the shape of the cavity can be effectively reconstructed with the NSNN.

Citation: Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu. A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28 (2) : 1123-1142. doi: 10.3934/era.2020062
##### References:
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Ma, The reciprocity gap functional method for the inverse scattering problem for cavities, Appl. Anal., 95 (2016), 1327-1346.  doi: 10.1080/00036811.2015.1064519.  Google Scholar [29] D. Xu, Z. Li and W. Wu, Convergence of gradient method for a fully recurrent neural network, Soft Computing, 14 (2010), 245-250.  doi: 10.1007/s00500-009-0398-0.  Google Scholar [30] D. Xu, Z. Li, W. Wu, X. Ding and D. Qu, Convergence of gradient descent algorithm for a recurrent neuron, International Symposium on Neural Networks, Springer, Berlin, Heidelberg, (2007), 117–122. doi: 10.1007/978-3-540-72395-0_16.  Google Scholar [31] W. Yin, W. Yang and H. Liu, A neural network scheme for recovering scattering obstacles with limited phaseless far-field data, Journal of Computational Physics, 417 (2020), 109594. doi: 10.1016/j.jcp.2020.109594.  Google Scholar [32] F. Zeng, P. Suarez and J. Sun, A decomposition method for an interior inverse scattering problem, Inverse Problems and Imaging, 7 (2013), 291-303.  doi: 10.3934/ipi.2013.7.291.  Google Scholar [33] D. Zhang, Y. Wang, Y. Guo and J. Li, Uniqueness in inverse cavity scattering problems with phaseless near-field data, Inverse Problems, 36 (2020), 025004. doi: 10.1088/1361-6420/ab53ee.  Google Scholar

show all references

##### References:
 [1] H. K. Aggarwal, M. P. Mani and M. Jacob, MoDL: Model based deep learning architecture for inverse problems, IEEE Transactions on Medical Imaging, 38 (2019), 394-405.  doi: 10.1109/TMI.2018.2865356.  Google Scholar [2] M. N. Akinci, Detection of the cavities inside a target with near field orthogonality sampling method, 2018 18th Mediterranean Microwave Symposium (MMS), IEEE, (2018), 391–393. doi: 10.1109/MMS.2018.8612107.  Google Scholar [3] G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 133 (2005), 1685-1691.  doi: 10.1090/S0002-9939-05-07810-X.  Google Scholar [4] E. Blåsten and H. Liu, On corners scattering stably, nearly non-scattering interrogating waves, and stable shape determination by a single far-field pattern, preprint, arXiv: 1611.03647. Google Scholar [5] E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, preprint, arXiv: 1705.00815. Google Scholar [6] F. Cakoni, D. Colton and S. Meng, The inverse scattering problem for a penetrable cavity with internal measurements, AMS Contemporary Mathematics, 615 (2014), 71-88.  doi: 10.13140/RG.2.1.1415.9520.  Google Scholar [7] J. Cheng and M. Yamamoto, Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves, Inverse Problems, 19 (2003), 1361-1384.  doi: 10.1088/0266-5611/19/6/008.  Google Scholar [8] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $4^{nd}$ edition, Applied Mathematical Sciences, 93. Springer, Cham, New York, 2019. doi: 10.1007/978-3-030-30351-8.  Google Scholar [9] L. Goodfellow, Y. Bengio and A. Courville, Deep Learning, MIT Press, London, 2017.   Google Scholar [10] J. Guo, Q. Yang, M. Cai, G. Yan and Z. Guo, Reconstruction of a crack with the incident waves and measurements inside a penetrable cavity, J. Inverse Ill-Posed Probl., 27 (2019), 643-656.  doi: 10.1515/jiip-2018-0023.  Google Scholar [11] Y. Hu, F. Cakoni and J. Liu, The inverse scattering problem for a partially coated cavity with interior measurements, Appl. Anal., 93 (2014), 936-956.  doi: 10.1080/00036811.2013.801458.  Google Scholar [12] P. Jakubik and R. Potthast, Testing the integrity of some cavity-the Cauchy problem and the range test, Appl. Numer. Math., 58 (2008), 899-914.  doi: 10.1016/j.apnum.2007.04.007.  Google Scholar [13] A. Karageorghis, D. Lesnic and L. Marin, The method of fundamental solutions for the identification of a scatterer with impedance boundary condition in interior inverse acoustic scattering, Eng. Anal. Bound. Elem., 92 (2018), 218-224.  doi: 10.1016/j.enganabound.2017.07.005.  Google Scholar [14] J. Li, H. Liu, W.-Y. Tsui and X. Wang, An inverse scattering approach for geometric body generation: A machine learning perspective, Mathematics in Engineering, 1 (2019), 800-823.  doi: 10.3934/mine.2019.4.800.  Google Scholar [15] P. Li and Y. Wang, Near-field imaging of interior cavities, Commun. Comput. Phys., 17 (2015), 542-563.  doi: 10.4208/cicp.010414.250914a.  Google Scholar [16] X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006, 18 pp. doi: 10.1088/0266-5611/30/1/015006.  Google Scholar [17] H. Liu, A global uniqueness for formally determined inverse electromagnetic obstacle scattering, Inverse Problems, 24 (2008), 035018, 13 pp. doi: 10.1088/0266-5611/24/3/035018.  Google Scholar [18] H. Liu, M. Petrini, L. Rondi and J. Xiao, Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements, J. Differential Equations, 262 (2017), 1631-1670.  doi: 10.1016/j.jde.2016.10.021.  Google Scholar [19] H. Liu, L. Rondi and J. Xiao, Mosco convergence for $H(\rm curl)$ spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems, J. Eur. Math. Soc. (JEMS), 21 (2019), 2945-2993.  doi: 10.4171/JEMS/895.  Google Scholar [20] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar [21] H. Liu and J. Zou, On unique determination of partially coated polyhedral scatterers with far field measurements, Inverse Problems, 23 (2007), 297-308.  doi: 10.1088/0266-5611/23/1/016.  Google Scholar [22] H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Problems, 27 (2011), 035005, 17 pp. doi: 10.1088/0266-5611/27/3/035005.  Google Scholar [23] H. Qin and D. Colton, The inverse scattering problem for cavities, Appl. Numer. Math., 62 (2012), 699-708.  doi: 10.1016/j.apnum.2010.10.011.  Google Scholar [24] H. Qin and D. Colton, The inverse scattering problem for cavities with impedance boundary condition, Adv. Comput. Math., 36 (2012), 157-174.  doi: 10.1007/s10444-011-9179-2.  Google Scholar [25] F. Qu, J. Yang and H. Zhang, Shape reconstruction in inverse scattering by an inhomogeneous cavity with internal measurements, SIAM J. Imaging Sci., 12 (2019), 788-808.  doi: 10.1137/18M1232401.  Google Scholar [26] L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement, Indiana Univ. Math. J., 57 (2008), 1377-1408.  doi: 10.1512/iumj.2008.57.3217.  Google Scholar [27] Y. Sanghvi, Y. Kalepu and U. K. Khankhoje, Embedding deep learning in inverse scattering problems, IEEE Transactions on Computational Imaging, 6 (2020), 46-56.  doi: 10.1109/TCI.2019.2915580.  Google Scholar [28] Y. Sun, Y. Guo and F. Ma, The reciprocity gap functional method for the inverse scattering problem for cavities, Appl. Anal., 95 (2016), 1327-1346.  doi: 10.1080/00036811.2015.1064519.  Google Scholar [29] D. Xu, Z. Li and W. Wu, Convergence of gradient method for a fully recurrent neural network, Soft Computing, 14 (2010), 245-250.  doi: 10.1007/s00500-009-0398-0.  Google Scholar [30] D. Xu, Z. Li, W. Wu, X. Ding and D. Qu, Convergence of gradient descent algorithm for a recurrent neuron, International Symposium on Neural Networks, Springer, Berlin, Heidelberg, (2007), 117–122. doi: 10.1007/978-3-540-72395-0_16.  Google Scholar [31] W. Yin, W. Yang and H. Liu, A neural network scheme for recovering scattering obstacles with limited phaseless far-field data, Journal of Computational Physics, 417 (2020), 109594. doi: 10.1016/j.jcp.2020.109594.  Google Scholar [32] F. Zeng, P. Suarez and J. Sun, A decomposition method for an interior inverse scattering problem, Inverse Problems and Imaging, 7 (2013), 291-303.  doi: 10.3934/ipi.2013.7.291.  Google Scholar [33] D. Zhang, Y. Wang, Y. Guo and J. Li, Uniqueness in inverse cavity scattering problems with phaseless near-field data, Inverse Problems, 36 (2020), 025004. doi: 10.1088/1361-6420/ab53ee.  Google Scholar
A schematic of the problem geometry
NSNN structure
Feedforward neural network structure
Reconstruct three shapes of cavity under the conditions of single point source incidence and limited observation points
Reconstruct the kite-shaped cavity with different noises
Reconstruct the peanut-shaped cavity with different noises
Reconstruct the starfish-shaped cavity with different noises
Reconstruct the kite-shaped cavity under different $m$
Reconstruct the kite-shaped cavity under different the limited-aperture range conditions
Parameter values of the NSNN
 Parameter value The Near-field shape of layer 2 Learning rate $\alpha$ 0.0001 Dropout 0.5 O 256 $\mathcal{O}$ 256 Batch 1000 Epoch: $t$ 100
 Parameter value The Near-field shape of layer 2 Learning rate $\alpha$ 0.0001 Dropout 0.5 O 256 $\mathcal{O}$ 256 Batch 1000 Epoch: $t$ 100
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