Vol. 108
Latest Volume
All Volumes
PIERC 142 [2024] PIERC 141 [2024] PIERC 140 [2024] PIERC 139 [2024] PIERC 138 [2023] PIERC 137 [2023] PIERC 136 [2023] PIERC 135 [2023] PIERC 134 [2023] PIERC 133 [2023] PIERC 132 [2023] PIERC 131 [2023] PIERC 130 [2023] PIERC 129 [2023] PIERC 128 [2023] PIERC 127 [2022] PIERC 126 [2022] PIERC 125 [2022] PIERC 124 [2022] PIERC 123 [2022] PIERC 122 [2022] PIERC 121 [2022] PIERC 120 [2022] PIERC 119 [2022] PIERC 118 [2022] PIERC 117 [2021] PIERC 116 [2021] PIERC 115 [2021] PIERC 114 [2021] PIERC 113 [2021] PIERC 112 [2021] PIERC 111 [2021] PIERC 110 [2021] PIERC 109 [2021] PIERC 108 [2021] PIERC 107 [2021] PIERC 106 [2020] PIERC 105 [2020] PIERC 104 [2020] PIERC 103 [2020] PIERC 102 [2020] PIERC 101 [2020] PIERC 100 [2020] PIERC 99 [2020] PIERC 98 [2020] PIERC 97 [2019] PIERC 96 [2019] PIERC 95 [2019] PIERC 94 [2019] PIERC 93 [2019] PIERC 92 [2019] PIERC 91 [2019] PIERC 90 [2019] PIERC 89 [2019] PIERC 88 [2018] PIERC 87 [2018] PIERC 86 [2018] PIERC 85 [2018] PIERC 84 [2018] PIERC 83 [2018] PIERC 82 [2018] PIERC 81 [2018] PIERC 80 [2018] PIERC 79 [2017] PIERC 78 [2017] PIERC 77 [2017] PIERC 76 [2017] PIERC 75 [2017] PIERC 74 [2017] PIERC 73 [2017] PIERC 72 [2017] PIERC 71 [2017] PIERC 70 [2016] PIERC 69 [2016] PIERC 68 [2016] PIERC 67 [2016] PIERC 66 [2016] PIERC 65 [2016] PIERC 64 [2016] PIERC 63 [2016] PIERC 62 [2016] PIERC 61 [2016] PIERC 60 [2015] PIERC 59 [2015] PIERC 58 [2015] PIERC 57 [2015] PIERC 56 [2015] PIERC 55 [2014] PIERC 54 [2014] PIERC 53 [2014] PIERC 52 [2014] PIERC 51 [2014] PIERC 50 [2014] PIERC 49 [2014] PIERC 48 [2014] PIERC 47 [2014] PIERC 46 [2014] PIERC 45 [2013] PIERC 44 [2013] PIERC 43 [2013] PIERC 42 [2013] PIERC 41 [2013] PIERC 40 [2013] PIERC 39 [2013] PIERC 38 [2013] PIERC 37 [2013] PIERC 36 [2013] PIERC 35 [2013] PIERC 34 [2013] PIERC 33 [2012] PIERC 32 [2012] PIERC 31 [2012] PIERC 30 [2012] PIERC 29 [2012] PIERC 28 [2012] PIERC 27 [2012] PIERC 26 [2012] PIERC 25 [2012] PIERC 24 [2011] PIERC 23 [2011] PIERC 22 [2011] PIERC 21 [2011] PIERC 20 [2011] PIERC 19 [2011] PIERC 18 [2011] PIERC 17 [2010] PIERC 16 [2010] PIERC 15 [2010] PIERC 14 [2010] PIERC 13 [2010] PIERC 12 [2010] PIERC 11 [2009] PIERC 10 [2009] PIERC 9 [2009] PIERC 8 [2009] PIERC 7 [2009] PIERC 6 [2009] PIERC 5 [2008] PIERC 4 [2008] PIERC 3 [2008] PIERC 2 [2008] PIERC 1 [2008]
2020-12-31
Data-Driven Identification of Governing Partial Differential Equations for the Transmission Line Systems
By
Progress In Electromagnetics Research C, Vol. 108, 23-36, 2021
Abstract
Discovering governing equations for transmission line is essential for the study on its properties, especially when the nonlinearity is introduced in a transmission line system. In this paper, we propose a novel data-driven approach for deriving the governing partial differential equations based on the spatial-temporal samples of current and voltage in the transmission line system. The proposed method is based on the ridge regression algorithm to determine the active spatial differential terms from the candidate library that includes nonlinear functions, in which the time and spatial derivatives are estimated by using polynomial interpolation. Three examples, including uniform and nonuniform transmission lines and a specific type of nonlinear transmission line for soliton generation, are provided to benchmark the performance of the proposed approach. The results demonstrate that the newly proposed approach can inverse the distributed circuit parameters and also discover the governing partial differential equations in the linear and nonlinear transmission line systems. Our proposed data-driven method for deriving governing equations could provide a practical tool in transmission line modeling.
Citation
Yanming Zhang, and Li Jun Jiang, "Data-Driven Identification of Governing Partial Differential Equations for the Transmission Line Systems," Progress In Electromagnetics Research C, Vol. 108, 23-36, 2021.
doi:10.2528/PIERC20102705
References

1. Stinehelfer, H. E., "An accurate calculation of uniform microstrip transmission lines," IEEE Trans. Microw. Theory Tech., Vol. 16, No. 7, 439-444, 1968.

2. Edward, G. C., "Theory and design of transmission line all-pass equalizers," IEEE Trans. Microw. Theory Tech., Vol. 17, No. 1, 28-38, 1969.

3. Kimionis, J., A. Collado, M. M. Tentzeris, and A. Georgiadis, "Octave and decade printed uwb rectifiers based on nonuniform transmission lines for energy harvesting," IEEE Trans. Microw. Theory Tech., Vol. 65, No. 11, 4326-4334, 2017.

4. Zhao, Y., S. Hemour, T. Liu, and K. Wu, "Nonuniformly distributed electronic impedance synthesizer," IEEE Trans. Microw. Theory Tech., Vol. 66, No. 11, 4883-4897, 2018.

5. Ramirez, A. I., A. Semlyen, and R. Iravani, "Modeling nonuniform transmission lines for time domain simulation of electromagnetic transients," IEEE Trans. Power Deliv., Vol. 18, No. 3, 968-974, 2003.

6. Lu, K., "An efficient method for analysis of arbitrary nonuniform transmission lines," IEEE Trans. Microw. Theory Tech., Vol. 45, No. 1, 9-14, 1997.

7. Nikoo, M. S. and S. M.-A. Hashemi, "New soliton solution of a varactor-loaded nonlinear transmission line," IEEE Trans. Microw. Theory Tech., Vol. 65, No. 11, 4084-4092, 2017.

8. Schleder, G. R., A. C. M. Padilha, C. M. Acosta, M. Costa, and A. Fazzio, "From DFT to machine learning: Recent approaches to materials science — A review," J. Phys. Materials, Vol. 2, No. 3, 032001, 2019.

9. Iten, R., T. Metger, H. Wilming, L. Del Rio, and R. Renner, "Discovering physical concepts with neural networks," Phys. Rev. Lett., Vol. 124, No. 1, 010508, 2020.

10. Sugihara, G., R. May, H. Ye, C.-H. Hsieh, E. Deyle, M. Fogarty, and S. Munch, "Detecting causality in complex ecosystems," Science, Vol. 338, No. 6106, 496-500, 2012.

11. Ye, H., R. J. Beamish, S. M. Glaser, S. C. H. Grant, C.-H. Hsieh, L. J. Richards, J. T. Schnute, and G. Sugihara, "Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling," Proc. Natl. Acad. Sci. U. S. A., Vol. 112, No. 13, E1569-E1576, 2015.

12. Kevrekidis, I. G., C. William Gear, J. M. Hyman, P. G. Kevrekidid, O. Runborg, C. Theodoropoulos, et al. "Equation-free, coarse-grained multiscale computation: Enabling mocroscopic simulators to perform system-level analysis," Commun. Math. Sci., Vol. 1, No. 4, 715-762, 2003.

13. Voss, H. U., P. Kolodner, M. Abel, and J. Kurths, "Amplitude equations from spatiotemporal binary-fluid convection data," Phys. Rev. Lett., Vol. 83, No. 17, 3422, 1999.

14. Guo, L. and S. A. Billings, "Identification of partial differential equation models for continuous spatio-temporal dynamical systems," IEEE Trans. Circuits Syst. II — Express Briefs, Vol. 53, No. 8, 657-661, 2006.

15. Guo, L. Z., S. A. Billings, and D. Coca, "Identification of partial differential equation models for a class of multiscale spatio-temporal dynamical systems," Int. J. Control, Vol. 83, No. 1, 40-48, 2010.

16. Gonzalez-Garcia, R., R. Rico-Martinez, and I. G. Kevrekidis, "Identification of distributed parameter systems: A neural net based approach," Comput. Chem. Eng., Vol. 22, S965-S968, 1998.

17. Giannakis, D. and A. J. Majda, "Nonlinear Laplacian spectral analysis for time series with intermittency and low-frequency variability," Proc. Natl. Acad. Sci. U. S. A., Vol. 109, No. 7, 2222-2227, 2012.

18. Roberts, A. J., "Model emergent dynamics in complex systems," SIAM, Vol. 20, 2014.

19. Majda, A. J., C. Franzke, and D. Crommelin, "Normal forms for reduced stochastic climate models," Proc. Natl. Acad. Sci. U. S. A., Vol. 106, No. 10, 3649-3653, 2009.

20. Daniels, B. C. and I. Nemenman, "Automated adaptive inference of phenomenological dynamical models," Nat. Commun., Vol. 6, No. 1, 1-8, 2015.

21. Bongard, J. and H. Lipson, "Automated reverse engineering of nonlinear dynamical systems," Proc. Natl. Acad. Sci. U. S. A., Vol. 104, No. 24, 9943-9948, 2007.

22. Schmidt, M. and H. Lipson, "Distilling free-form natural laws from experimental data," Science, Vol. 324, No. 5923, 81-85, 2009.

23. Raissi, M. and G. E. Karniadakis, "Hidden physics models: Machine learning of nonlinear partial differential equations," J. Comput. Phys., Vol. 357, 125-141, 2018.

24. Raissi, M., P. Perdikaris, and G. E. Karniadakis, "Physics informed deep learning (part II, Data-driven discovery of nonlinear partial differential equations,", arxiv. arXiv preprint arXiv:1711.10561, 2017.

25. Long, Z., Y. Lu, and B. Dong, "PDE-Net 2.0: Learning pdes from data with a numericsymbolic hybrid deep network," J. Comput. Phys., Vol. 399, 108925, 2019.

26. Brunton, S. L., J. L. Proctor, and J. N. Kutz, "Discovering governing equations from data by sparse identification of nonlinear dynamical systems," Proc. Natl. Acad. Sci. U. S. A., Vol. 113, No. 15, 3932-3937, 2016.

27. Schaeffer, H., "Learning partial differential equations via data discovery and sparse optimization," Proc. R. Soc. A — Math. Phys. Eng. Sci., Vol. 473, No. 2197, 20160446, 2017.

28. Rudy, S. H., S. L. Brunton, J. L. Proctor, and J. N. Kutz, "Data-driven discovery of partial differential equations," Sci. Adv., Vol. 3, No. 4, e1602614, 2017.

29. Champion, K., B. Lusch, J. N. Kutz, and S. L. Brunton, "Data-driven discovery of coordinates and governing equations," Proc. Natl. Acad. Sci. U. S. A., Vol. 116, No. 45, 22445-22451, 2019.

30. Mangan, N. M., S. L. Brunton, J. L. Proctor, and J. N. Kutz, "Inferring biological networks by sparse identification of nonlinear dynamics," IEEE Trans. Mol. Biol. Multi-Scale Commun., Vol. 2, No. 1, 52-63, 2016.

31. Himanen, L., A. Geurts, A. S. Foster, and P. Rinke, "Data-driven materials science: Status, challenges, and perspectives," Adv. Sci., Vol. 6, No. 21, 1900808, 2019.

32. Murphy, K. P., Machine Learning: A Probabilistic Perspective, MIT Press, 2012.

33. Hoerl, A. E. and R. W. Kennard, "Ridge regression: Biased estimation for nonorthogonal problems," Technometrics, Vol. 12, No. 1, 55-67, 1970.

34. LeVeque, R. J., "Finite difference methods for ordinary and partial differential equations: Steady-state and time-dependent problems," SIAM, Vol. 98, 2007.

35. Nathan Kutz, J., Data-driven Modeling & Scientific Computation: Methods for Complex Systems & Big Data, Oxford University Press, 2013.

36. Elsherbeni, A. Z., V. Demir, et al. The Finite-Difference Time-Domain Method for Electromagnetics with MATLAB Simulations, SciTech Pub., 2009.

37. Knowles, I. and R. J. Renka, "Methods for numerical differentiation of noisy data," Electron. J. Differ. Equ., Vol. 21, 235-246, 2014.

38. Bruno, O. and D. Hoch, "Numerical differentiation of approximated functions with limited order-ofaccuracy deterioration," SIAM J. Numer. Anal., Vol. 50, No. 3, 1581-1603, 2012.

39. Nevels, R. and J. Miller, "A simple equation for analysis of nonuniform transmission lines," IEEE Trans. Microw. Theory Tech., Vol. 49, No. 4, 721-724, 2001.

40. Watanabe, K., T. Sekine, and Y. Takahashi, "A FDTD method for nonuniform transmission line analysis using Yee’s-lattice and wavelet expansion," 2009 IEEE MTT-S International Microwave Workshop Series on Signal Integrity and High-Speed Interconnects, 83-86, 2009.

41. Sebastiano, G. S., P. Pantano, and P. Tucci, "An electrical model for the Korteweg-de Vries equation," Am. J. Phys., Vol. 52, No. 3, 238-243, 1984.

42. Ludu, A., Nonlinear Waves and Solitons on Contours and Closed Surfaces, Springer Science & Business Media, 2012.

43. Darling, J. D. C. and P. W. Smith, "High-power pulsed RF extraction from nonlinear lumped element transmission lines," IEEE Trans. Plasma Sci., Vol. 36, No. 5, 2598-2603, 2008.

44. Kuek, N. S., A. C. Liew, E. Schamiloglu, and J. O. Rossi, "Circuit modeling of nonlinear lumped element transmission lines including hybrid lines," IEEE Trans. Plasma Sci., Vol. 40, No. 10, 2523-2534, 2012.

45. Ricketts, D. S., X. Li, M. DePetro, and D. Ham, "A self-sustained electrical soliton oscillator," IEEE MTT-S International Microwave Symposium Digest, 4, IEEE, 2005.

46. Raissi, M., A. Yazdani, and G. E. Karniadakis, "Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations," Science, Vol. 367, No. 6481, 1026-1030, 2020.