DYNAMICAL SYSTEMS MODELING BASED ON POLYNOMIAL NEURAL NETWORKS

In the article, a polynomial neural network architecture is presented. This architecture is utilized for dynamical systems identification. The given approach is based on matrix representation of Lie transform, that is useful for investigation of nonlinear systems of ordinary differential equations. The polynomial neural network, in this case, can play a role of an effective and efficient method of investigation of dynamical systems. Moreover, it joints advantages of parallel computing architecture with the strong mathematical theory of differential equations. The key concepts and formulations are briefly described. The numerical matrix integration of the systems of differential equations is also presented. As an example, the identification of the simple model problem is considered as well as an application of the technique for modeling of vessel motion is presented. In the conclusion the limitations and further development of the method is indicated.

1. Mall S., Chakraverty S. Comparison of artificial neural network architecture in solving ordinary differential equations // Advances in Artificial Neural Systems, 2013.

2. Lagaris I. E., Likas A., Fotiadis D. I. Artificial neural networks for solving ordinary and partial differential equations // Technical report, 1997. URL https: //arxiv.org/pdf/physics/9705023.pdf.

3. Lapedes A., Farber R. Nonlinear signal processing using neural networks: Prediction and system modelling // IEEE International Conference on Neural Networks, San Diego, CA, USA, 1987.

4. Lewis F. L., Ge S. S. Neural Networks in Feedback Control Systems, in Mechanical Engineers // Handbook: Instrumentation, Systems, Controls, and MEMS, volume 2. John Wiley and Sons, Hoboken, NJ, USA, third edition (ed m. kutz) edition, 2005

5. Chen S., Billings S. A. Neural networks for nonlinear dynamic system modelling and identification // International Journal of Control, 56(2):319–346, 1992.

6. Cybenko, G. V. Approximation by Superpositions of a Sigmoidal function // Mathematics of Control Signals and Systems,V. 2, P. 303-314, 1989.

7. Dragt A. Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics. 2011. URL http://inspirehep.net/record/955313/files/TOC28Nov2011.pdf.

8. Andrianov S. N. Modeling of Beam Dynamics Control Systems. SaintPetersburg, 2004. 368 с.

9. Ivanov A. N., Kuznetcov P.M. Identification of dynamical systems based on nonlinear matrix Lie transform // Vestnik UGATU. — 2014. — Т. 18. — No 2(63). — C. 251–256.

10. E. Tu, G. Zhang, L. Rachmawati, E. Rajabally and G. B. Huang, Exploiting AIS Data for Intelligent Maritime Navigation: A Comprehensive Survey From Data to Methodology // IEEE Transactions on Intelligent Transportation Systems, vol. PP, no. 99, pp. 1-24.

11. Sholokhova A.A. Anomaly detection in Sensor Data in Application to the Analysis of Maritime Vessels Motion // Journ. of Modelling, Optimization and Information Technology. — 2017. — № 3(18). — 19 С.