volume | PIER Journals

Abstract

Coarse discretization introduces significant errors in the solution of scattering problems, in part due to discretization errors in the contrast operator. We present a procedure for the automatic construction of a modified contrast operator for electromagnetic scattering problems by using trainable neural networks to represent a modified contrast operator. We achieve a higher accuracy on a coarse discretization while still keeping computation time down compared to a fine discretization. By using synthetic data from a full-wave Maxwell solver to train the network for one-dimensional slab scatterers and two-dimensional polygonal scatterers, we are able to use the techniques found in deep learning to improve accuracy in coarse-grid forward scattering problems.

1. Gibson, Walton C., The Method of Moments in Electromagnetics, CRC Press, 2021.
doi:10.1201/9780429355509

2. Van der Vorst, H. A., "Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems," SIAM Journal on Scientific and Statistical Computing, Vol. 13, No. 2, 631-644, 1992.
doi:10.1137/0913035

3. Saad, Youcef and Martin H. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM Journal on Scientific and Statistical Computing, Vol. 7, No. 3, 856-869, 1986.
doi:10.1137/0907058

4. Zwamborn, P. and P. M. van den Berg, "The three dimensional weak form of the conjugate gradient FFT method for solving scattering problems," IEEE Transactions on Microwave Theory and Techniques, Vol. 40, No. 9, 1757-1766, 1992.
doi:10.1109/22.156602

5. Dilz, Roeland Johannes and Martijn Constant van Beurden, "An efficient spatial spectral integral-equation method for EM scattering from finite objects in layered media," 2016 International Conference on Electromagnetics in Advanced Applications (ICEAA), 509-511, Cairns, Australia, Sep. 2016.
doi:10.1109/iceaa.2016.7731441

6. Dilz, R. J. and M. C. van Beurden, "A domain integral equation approach for simulating two dimensional transverse electric scattering in a layered medium with a Gabor frame discretization," Journal of Computational Physics, Vol. 345, 528-542, 2017.
doi:10.1016/j.jcp.2017.05.034

7. Dilz, Roeland J., Mark G. M. M. van Kraaij, and Martijn C. van Beurden, "A 3D spatial spectral integral equation method for electromagnetic scattering from finite objects in a layered medium," Optical and Quantum Electronics, Vol. 50, No. 5, 206, 2018.
doi:10.1007/s11082-018-1471-7

8. Eijsvogel, S., R. J. Dilz, R. Bojanić, and M. C. van Beurden, "Phaseless inverse scattering with a parametrized spatial spectral volume integral equation for finite scatterers in the soft X-ray regime," Journal of the Optical Society of America A, Vol. 41, No. 11, 2076-2089, 2024.
doi:10.1364/josaa.515382

9. Ergul, Ozgur and Levent Gurel, The Multilevel Fast Multipole Algorithm (MLFMA) for Solving Large-Scale Computational Electromagnetics Problems, John Wiley & Sons, 2014.

10. Hu, Bin, Weng Cho Chew, Eric Michielssen, and Junsheng Zhao, "Fast inhomogeneous plane wave algorithm for the fast analysis of two-dimensional scattering problems," Radio Science, Vol. 34, No. 4, 759-772, 1999.
doi:10.1029/1999rs900038

11. Chen, Yongpin, Jun Hu, Zaiping Nie, and Lin Lei, "A FAFFA-FIPWA algorithm for two-dimensional electromagnetic scattering problem," 2005 IEEE International Symposium on Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications, Vol. 1, 479-482, Beijing, China, 2005.
doi:10.1109/mape.2005.1617953

12. Weiss, Thomas, Gérard Granet, Nikolay A. Gippius, Sergei G. Tikhodeev, and Harald Giessen, "Matched coordinates and adaptive spatial resolution in the Fourier modal method," Optics Express, Vol. 17, No. 10, 8051-8061, 2009.
doi:10.1364/oe.17.008051

13. Krizhevsky, Alex, Ilya Sutskever, and Geoffrey E. Hinton, "Imagenet classification with deep convolutional neural networks," Advances in Neural Information Processing Systems, Vol. 25, 2012.

14. He, Kaiming, Xiangyu Zhang, Shaoqing Ren, and Jian Sun, "Deep residual learning for image recognition," arXiv preprint arXiv:1512.03385, 2015.
doi:10.48550/arXiv.1512.03385

15. Ronneberger, Olaf, Philipp Fischer, and Thomas Brox, "U-Net: Convolutional networks for biomedical image segmentation," arXiv preprint arXiv:1505.04597, 2015.
doi:10.48550/arXiv.1505.04597

16. Hinton, Geoffrey, Li Deng, Dong Yu, George E. Dahl, Abdel-Rahman Mohamed, Navdeep Jaitly, Andrew Senior, Vincent Vanhoucke, Patrick Nguyen, Tara N. Sainath, and Brian Kingsbury, "Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups," IEEE Signal Processing Magazine, Vol. 29, No. 6, 82-97, 2012.
doi:10.1109/msp.2012.2205597

17. Vaswani, Ashish, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin, "Attention is all you need," arXiv preprint arXiv:1706.03762, 2017.
doi:10.48550/arXiv.1706.03762

18. Raissi, Maziar, Paris Perdikaris, and George Em Karniadakis, "Physics informed deep learning (Part I): Data-driven solutions of nonlinear partial differential equations," arXiv preprint arXiv:1711.10561, 2017.
doi:10.48550/arXiv.1711.10561

19. Guo, Liangshuai, Maokun Li, Shenheng Xu, and Fan Yang, "Study on a recurrent convolutional neural network based FDTD method," 2019 International Applied Computational Electromagnetics Society Symposium --- China (ACES), Vol. 1, 1-2, Nanjing, China, 2019.
doi:10.23919/aces48530.2019.9060707

20. Yao, He Ming and Li Jun Jiang, "Machine learning based neural network solving methods for the FDTD method," 2018 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, 2321-2322, Boston, MA, USA, 2018.
doi:10.1109/apusncursinrsm.2018.8608745

21. Yee, Kane, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Transactions on Antennas and Propagation, Vol. 14, No. 3, 302-307, 1966.
doi:10.1109/tap.1966.1138693

22. LeCun, Yann, Yoshua Bengio, and Geoffrey Hinton, "Deep learning," Nature, Vol. 521, No. 7553, 436-444, 2015.
doi:10.1038/nature14539

23. Rumelhart, D. E. and J. L. McClelland, Learning Internal Representations by Error Propagation, 318-362, MIT Press, 1987.
doi:10.1016/b978-1-4832-1446-7.50035-2

24. Sak, Haşim, Andrew Senior, and Françoise Beaufays, "Long short-term memory recurrent neural network architectures for large scale acoustic modeling," Proceedings of the Annual Conference of the International Speech Communication Association, 338-342, Singapore, 2014.
doi:10.21437/Interspeech.2014-80

25. Yao, He Ming and Lijun Jiang, "Machine-learning-based PML for the FDTD method," IEEE Antennas and Wireless Propagation Letters, Vol. 18, No. 1, 192-196, 2018.
doi:10.1109/lawp.2018.2885570

26. Key, Cam and Branislav M. Notaroš, "Data-enabled advancement of computation in engineering: A robust machine learning approach to accelerating variational methods in electromagnetics and other disciplines," IEEE Antennas and Wireless Propagation Letters, Vol. 19, No. 4, 626-630, 2020.
doi:10.1109/lawp.2020.2973937

27. Jin, Jian-Ming, The Finite Element Method in Electromagnetics, John Wiley & Sons, 2002.

28. Key, Cam and Branislav M. Notaroš, "Predicting macro basis functions for method of moments scattering problems using deep neural networks," IEEE Antennas and Wireless Propagation Letters, Vol. 20, No. 7, 1200-1204, 2021.
doi:10.1109/lawp.2021.3075370

29. Guo, Rui, Tao Shan, Xiaoqian Song, Maokun Li, Fan Yang, Shenheng Xu, and Aria Abubakar, "Physics embedded deep neural network for solving volume integral equation: 2-D case," IEEE Transactions on Antennas and Propagation, Vol. 70, No. 8, 6135-6147, 2022.
doi:10.1109/tap.2021.3070152

30. Guo, Rui, Zhichao Lin, Tao Shan, Xiaoqian Song, Maokun Li, Fan Yang, Shenheng Xu, and Aria Abubakar, "Physics embedded deep neural network for solving full-wave inverse scattering problems," IEEE Transactions on Antennas and Propagation, Vol. 70, No. 8, 6148-6159, 2022.
doi:10.1109/tap.2021.3102135

31. Xiao, Li-Ye, Jun-Nan Yi, Yiqian Mao, Xin-Yue Qi, Ronghan Hong, and Qing Huo Liu, "A novel optical proximity correction (OPC) system based on deep learning method for the extreme ultraviolet (EUV) lithography," Progress In Electromagnetics Research, Vol. 176, 95-108, 2023.
doi:10.2528/pier22101601

32. Xu, Feng and Shilei Fu, "Modeling EM problem with deep neural networks," 2018 IEEE International Conference on Computational Electromagnetics (ICCEM), 1-2, Chengdu, China, 2018.
doi:10.1109/compem.2018.8496532

33. Van den Hof, Daan, Martijn C. van Beurden, and Roeland Dilz, "Flexible discretization of singular Green functions using a composite spectral integration path," Progress In Electromagnetics Research B, Vol. 107, 77-90, 2024.
doi:10.2528/pierb24051404

34. Abadi, M., A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, "TensorFlow: Large-scale machine learning on heterogeneous systems," arXiv:1603.04467, 2016.
doi:10.48550/arXiv.1603.04467

35. Dilz, Roeland, "The windowed Fourier series with a complex-plane path deformation as an efficient discretization for spatial spectral integral equations," AES 2024 Rome-Italy: The 10th International Conference on Antennas and Electromagnetic Systems, 324-325, Rome, Italy, Jun. 2024.

36. Ramachandran, Prajit, Barret Zoph, and Quoc V. Le, "Searching for activation functions," arXiv preprint arXiv:1710.05941, 2017.
doi:10.48550/arXiv.1710.05941

37. Harris, Charles R., K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, et al. "Array programming with NumPy," Nature, Vol. 585, No. 7825, 357-362, Sep. 2020.
doi:10.1038/s41586-020-2649-2

38. Hunter, John D., "Matplotlib: A 2D graphics environment," Computing in Science & Engineering, Vol. 9, No. 3, 90-95, 2007.
doi:10.1109/mcse.2007.55

39. Eijsvogel, S., L. Sun, F. Sepehripour, R. J. Dilz, and M. C. van Beurden, "Describing discontinuous finite 3D scattering objects in Gabor coefficients: Fast and accurate methods," Journal of the Optical Society of America A, Vol. 39, No. 1, 86-97, 2022.
doi:10.1364/josaa.438866

40. Van Der Sijs, T. A., O. El Gawhary, and H. P. Urbach, "Electromagnetic scattering beyond the weak regime: Solving the problem of divergent Born perturbation series by Padé approximants," Physical Review Research, Vol. 2, No. 1, 013308, 2020.
doi:10.1103/physrevresearch.2.013308

41. Kingma, Diederik P. and Jimmy Ba, "Adam: A method for stochastic optimization," arXiv preprint arXiv:1412.6980, Vol. 1412, No. 6, 2014.
doi:10.48550/arXiv.1412.6980

Mr. Daan van den Hof

Prof. Martijn Constant van Beurden

Dr. Roeland J. Dilz