Skip to main content
SpringerLink
Log in

Theoretical Analysis of Gradient-Zhang Neural Network for Time-Varying Equations and Improved Method for Linear Equations

  • Conference paper
  • First Online:
Neural Information Processing (ICONIP 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14447))

Included in the following conference series:

  • 652 Accesses

Abstract

Solving time-varying equations is fundamental in science and engineering. This paper aims to find a fast-converging and high-precision method for solving time-varying equations. We combine two classes of feedback neural networks, i.e., gradient neural network (GNN) and Zhang neural network (ZNN), to construct a continuous gradient-Zhang neural network (GZNN) model. Our research shows that GZNN has the advantages of high convergence precision of ZNN and fast convergence speed of GNN in certain cases, i.e., all the eigenvalues of Jacobian matrix of the time-varying equations multiplied by its transpose are larger than 1. Furthermore, we conduct the different detailed mathematical proof and theoretical analysis to establish the stability and convergence of the GZNN model. Additionally, we discretize the GZNN model by utilizing time discretization formulas (i.e., Euler and Taylor-Zhang discretization formulas), to construct corresponding discrete GZNN algorithms for solving discrete time-varying problems. Different discretization formulas can construct discrete algorithms with varying precision. As the number of time sampling instants increases, the precision of discrete algorithms can be further improved. Furthermore, we improve the matrix inverse operation in the GZNN model and develop inverse-free GZNN algorithms to solve linear problems, effectively reducing their time complexity. Finally, numerical experiments are conducted to validate the feasibility of GZNN model and the corresponding discrete algorithms in solving time-varying equations, as well as the efficiency of the inverse-free method in solving linear equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Chan, P.K., Chen, D.Y.: A CMOS ISFET interface circuit with dynamic current temperature compensation technique. IEEE Trans. Circuits Syst. I Regul. Pap. 54(1), 119–129 (2007)

    Article  Google Scholar 

  2. Guo, D., Zhang, Y.: Zhang neural network, Getz-Marsden dynamic system, and discrete-time algorithms for time-varying matrix inversion with application to robots’ kinematic control. Neurocomputing 97, 22–32 (2012)

    Article  Google Scholar 

  3. Hassoun, M.H.: Fundamentals of Artificial Neural Networks. MIT Press, Cambridge (1995)

    Google Scholar 

  4. Hildebrand, F.B.: Introduction to Numerical Analysis. Courier Corporation, North Chelmsford (1987)

    Google Scholar 

  5. Jin, L., Li, S., Liao, B., Zhang, Z.: Zeroing neural networks: a survey. Neurocomputing 267, 597–604 (2017)

    Article  Google Scholar 

  6. Jin, L., Zhang, Y., Li, S., Zhang, Y.: Modified ZNN for time-varying quadratic programming with inherent tolerance to noises and its application to kinematic redundancy resolution of robot manipulators. IEEE Trans. Ind. Electron. 63(11), 6978–6988 (2016)

    Article  Google Scholar 

  7. Li, J., Wu, G., Li, C., Xiao, M., Zhang, Y.: GMDS-ZNN variants having errors proportional to sampling gap as compared with models 1 and 2 having higher precision. In: Proceedings of International Conference on Systems and Informatics, pp. 728–733 (2018)

    Google Scholar 

  8. Li, J., Zhang, Y., Mao, M.: Five-instant type discrete-time ZND solving discrete time-varying linear system, division and quadratic programming. Neurocomputing 331, 323–335 (2019)

    Article  Google Scholar 

  9. Li, S., Li, Y.: Nonlinearly activated neural network for solving time-varying complex Sylvester equation. IEEE Trans. Cybern. 44(8), 1397–1407 (2014)

    Article  Google Scholar 

  10. Liao, B., Zhang, Y.: Different complex ZFs leading to different complex ZNN models for time-varying complex generalized inverse matrices. IEEE Trans. Neural Netw. Learn. Syst. 25(9), 1621–1631 (2014)

    Article  Google Scholar 

  11. Oppenheim, A.V., Willsky, A.S., Nawab, S.H., Ding, J.J.: Signals and Systems. Prentice Hall, New Jersey (1997)

    Google Scholar 

  12. Qiu, B., Guo, J., Li, X., Zhang, Z., Zhang, Y.: Discrete-time advanced zeroing neurodynamic algorithm applied to future equality-constrained nonlinear optimization with various noises. IEEE Trans. Cybern. 52(5), 3539–3552 (2022)

    Article  Google Scholar 

  13. Sun, C., Ye, M., Hu, G.: Distributed time-varying quadratic optimization for multiple agents under undirected graphs. IEEE Trans. Autom. Control 62(7), 3687–3694 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wu, D., Zhang, Y., Guo, J., Li, Z., Ming, L.: GMDS-ZNN model 3 and its ten-instant discrete algorithm for time-variant matrix inversion compared with other multiple-instant ones. IEEE Access 8, 228188–228198 (2020)

    Article  Google Scholar 

  15. Yang, M., Zhang, Y., Hu, H.: Inverse-free DZNN models for solving time-dependent linear system via high-precision linear six-step method. IEEE Trans. Neural Netw. Learn. Syst. 1–12 (2022)

    Google Scholar 

  16. Yang, M., Zhang, Y., Zhang, Z., Hu, H.: Adaptive discrete ZND models for tracking control of redundant manipulator. IEEE Trans. Ind. Inf. 16(12), 7360–7368 (2020)

    Article  Google Scholar 

  17. Yu, H., Sung, Y.: Least squares approach to joint beam design for interference alignment in multiuser multi-input multi-output interference channels. IEEE Trans. Signal Process. 58(9), 4960–4966 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang, Y.: Analysis and design of recurrent neural networks and their applications to control and robotic systems. The Chinese University of Hong Kong (2003)

    Google Scholar 

  19. Zhang, Y., Ge, S.: Design and analysis of a general recurrent neural network model for time-varying matrix inversion. IEEE Trans. Neural Netw. 16(6), 1477–1490 (2005)

    Article  Google Scholar 

  20. Zhang, Y., Gong, H., Yang, M., Li, J., Yang, X.: Stepsize range and optimal value for Taylor-Zhang discretization formula applied to zeroing neurodynamics illustrated via future equality-constrained quadratic programming. IEEE Trans. Neural Netw. Learn. Syst. 30(3), 959–966 (2019)

    Article  MathSciNet  Google Scholar 

  21. Zhang, Y., Li, Z.: Zhang neural network for online solution of time-varying convex quadratic program subject to time-varying linear-equality constraints. Phys. Lett. A 373(18), 1639–1643 (2009)

    Article  MATH  Google Scholar 

  22. Zhang, Y., Ling, Y., Yang, M., Yang, S., Zhang, Z.: Inverse-free discrete ZNN models solving for future matrix pseudoinverse via combination of extrapolation and ZeaD formulas. IEEE Trans. Neural Netw. Learn. Syst. 32(6), 2663–2675 (2021)

    Article  MathSciNet  Google Scholar 

  23. Zhang, Y., Wang, C.: Gradient-Zhang neural network solving linear time-varying equations. In: Proceedings of IEEE Conference on Industrial Electronics and Applications, pp. 396–403 (2022)

    Google Scholar 

  24. Zhang, Y., Wang, J., Xia, Y.: A dual neural network for redundancy resolution of kinematically redundant manipulators subject to joint limits and joint velocity limits. IEEE Trans. Neural Netw. 14(3), 658–667 (2003)

    Article  Google Scholar 

  25. Zhang, Y., Wu, G., Qiu, B., Li, W., He, P.: Euler-discretized GZ-type complex neuronet computing real-time varying complex matrix inverse. In: Proceedings of Chinese Control Conference, pp. 3914–3919 (2017)

    Google Scholar 

  26. Zhang, Y., Xie, Y., Tan, H.: Time-varying Moore-Penrose inverse solving shows different Zhang functions leading to different ZNN models. In: Proceedings of Advances in Neural Networks - ISNN 2012, pp. 98–105 (2012)

    Google Scholar 

  27. Zhang, Y., Yi, C., Guo, D., Zheng, J.: Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation. Neural Comput. Appl. 20, 1–7 (2011)

    Article  Google Scholar 

  28. Zhang, Y., Yi, C., Ma, W.: Comparison on gradient-based neural dynamics and Zhang neural dynamics for online solution of nonlinear equations. In: Proceedings of International Symposium on Advances in Computation and Intelligence, pp. 269–279 (2008)

    Google Scholar 

Download references

Acknowledgements

This work is aided by the National Natural Science Foundation of China (with number 61976230), the Project Supported by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme (with number 2018), and the Key-Area Research and Development Program of Guangzhou (with number 202007030004), with corresponding author Yunong Zhang.

Author information

Authors and Affiliations

  1. School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou, 510006, People’s Republic of China

    Changyuan Wang & Yunong Zhang

Authors
  1. Changyuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunong Zhang .

Editor information

Editors and Affiliations

  1. Central South University, Changsha, China

    Biao Luo

  2. Chinese Academy of Sciences, Beijing, China

    Long Cheng

  3. Zhejiang University, Hangzhou, China

    Zheng-Guang Wu

  4. Guangdong University of Technology, Guangzhou, China

    Hongyi Li

  5. UNSW Sydney, Sydney, NSW, Australia

    Chaojie Li

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Wang, C., Zhang, Y. (2024). Theoretical Analysis of Gradient-Zhang Neural Network for Time-Varying Equations and Improved Method for Linear Equations. In: Luo, B., Cheng, L., Wu, ZG., Li, H., Li, C. (eds) Neural Information Processing. ICONIP 2023. Lecture Notes in Computer Science, vol 14447. Springer, Singapore. https://doi.org/10.1007/978-981-99-8079-6_22

Download citation

  • DOI: https://doi.org/10.1007/978-981-99-8079-6_22

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-99-8078-9

  • Online ISBN: 978-981-99-8079-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation