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[J]. Electronic Research Archive, 2020, 28(2): 1123-1142. doi: 10.3934/era.2020062
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.
[1] | MoDL: Model based deep learning architecture for inverse problems. IEEE Transactions on Medical Imaging (2019) 38: 394-405. |
[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 |
[3] | Determining a sound-soft polyhedral scatterer by a single far-field measurement. Proc. Amer. Math. Soc. (2005) 133: 1685-1691. |
[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. |
[5] | E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, preprint, arXiv: 1705.00815. |
[6] | The inverse scattering problem for a penetrable cavity with internal measurements. AMS Contemporary Mathematics (2014) 615: 71-88. |
[7] | Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves. Inverse Problems (2003) 19: 1361-1384. |
[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 |
[9] | (2017) Deep Learning.MIT Press. |
[10] | Reconstruction of a crack with the incident waves and measurements inside a penetrable cavity. J. Inverse Ill-Posed Probl. (2019) 27: 643-656. |
[11] | The inverse scattering problem for a partially coated cavity with interior measurements. Appl. Anal. (2014) 93: 936-956. |
[12] | Testing the integrity of some cavity-the Cauchy problem and the range test. Appl. Numer. Math. (2008) 58: 899-914. |
[13] | The method of fundamental solutions for the identification of a scatterer with impedance boundary condition in interior inverse acoustic scattering. Eng. Anal. Bound. Elem. (2018) 92: 218-224. |
[14] | An inverse scattering approach for geometric body generation: A machine learning perspective. Mathematics in Engineering (2019) 1: 800-823. |
[15] | Near-field imaging of interior cavities. Commun. Comput. Phys. (2015) 17: 542-563. |
[16] | X. Liu, The factorization method for cavities, Inverse Problems, 30 (2014), 015006, 18 pp. doi: 10.1088/0266-5611/30/1/015006 |
[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 |
[18] | Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements. J. Differential Equations (2017) 262: 1631-1670. |
[19] | 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) (2019) 21: 2945-2993. |
[20] | Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers. Inverse Problems (2006) 22: 515-524. |
[21] | On unique determination of partially coated polyhedral scatterers with far field measurements. Inverse Problems (2007) 23: 297-308. |
[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 |
[23] | The inverse scattering problem for cavities. Appl. Numer. Math. (2012) 62: 699-708. |
[24] | The inverse scattering problem for cavities with impedance boundary condition. Adv. Comput. Math. (2012) 36: 157-174. |
[25] | Shape reconstruction in inverse scattering by an inhomogeneous cavity with internal measurements. SIAM J. Imaging Sci. (2019) 12: 788-808. |
[26] | Stable determination of sound-soft polyhedral scatterers by a single measurement. Indiana Univ. Math. J. (2008) 57: 1377-1408. |
[27] | Embedding deep learning in inverse scattering problems. IEEE Transactions on Computational Imaging (2020) 6: 46-56. |
[28] | The reciprocity gap functional method for the inverse scattering problem for cavities. Appl. Anal. (2016) 95: 1327-1346. |
[29] | Convergence of gradient method for a fully recurrent neural network. Soft Computing (2010) 14: 245-250. |
[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 |
[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 |
[32] | A decomposition method for an interior inverse scattering problem. Inverse Problems and Imaging (2013) 7: 291-303. |
[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 |