volume | PIER Journals

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.

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

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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

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. Google Scholar

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

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. Google Scholar

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

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

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. Google Scholar

Mr. Yanming Zhang

Dr. Li Jun Jiang