Abstract
Differential equations can describe physical and natural systems, which behavior only explicit exact functions are not able to model. Complex dynamic systems are characterized by a high variability of time-fluctuating data relations of a great number of state variables. Systems of differential equations can describe them but they are too unstable to be modeled unambiguously by means of standard soft computing techniques. In some cases the correct form of a differential equation might absent or it is difficult to express. Differential polynomial neural network is a new neural network type, which forms and solves an unknown general partial differential equation of an approximation of a searched function, described by discrete data observations. It generates convergent sum series of relative partial polynomial derivative terms, which can substitute for a partial or/and ordinary differential equation solution. This type of non-linear regression decomposes a system model, described by the general differential equation, into low order composite partial polynomial fractions in an additive series solution. The differential network can model the dynamics of the complex weather system, using only several input variables in some cases. Comparisons were done with the recurrent neural network, often applied for simple and solid time-series models.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
National Weather service (NWS) local observations. www.wrh.noaa.gov/mesowest/getobext.php?wfo=tfx&sid=KHLN&num=168&raw=0&dbn=m&banner=header.
- 2.
National Climatic Data Center of National Oceanic and Atmospheric Administration (NOAA). http://cdo.ncdc.noaa.gov/qclcd_ascii/.
- 3.
Monthly station normals. www.ndsu.edu/ndsco/normals/7100normals/MTnorm.pdf.
References
Allahviranloo, T., Ahmady, E., Ahmady, N.: Nth-order fuzzy linear differential equations. Inf. Sci. 178, 1309–1324 (2008)
Bertsimas, D., Tsitsiklis, J.: Simulated annealing. Stat. Sci. 8(1), 10–15 (1993)
Chan, K., Chau, W.Y.: Mathematical theory of reduction of physical parameters and similarity analysis. Int. J. Theor. Phys. 18, 835–844 (1979)
Chaquet, J., Carmona, E.: Solving differential equations with fourier series and evolution strategies. Appl. Soft. Comput. 12, 3051–3062 (2012)
Chen, Y., Yang, B., Meng, Q., Zhao, Y., Abraham, A.: Time-series forecasting using a system of ordinary differential equations. Inf. Sci. 181, 106–114 (2011)
Iba, H.: Inference of differential equation models by genetic programming. Inf. Sci. 178(4), 4453–4468 (2008)
Ivakhnenko, A.: Polynomial theory of complex systems. IEEE Trans. Syst. 1, 4 (1971)
Nikolaev, N.Y., Iba, H.: Polynomial harmonic GMDH learning networks for time series modeling. Neural Networks 16, 1527–1540 (2003)
Nikolaev, N.Y., Iba, H.: Adaptive Learning of Polynomial Networks. Genetic and evolutionary computation series. Springer, New York (2006)
Tsoulos, I., Gavrilis, D., Glavas, E.: Solving differential equations with constructed neural networks. Neurocomputing 72, 2385–2391 (2009)
Zjavka, L.: Recognition of generalized patterns by a differential polynomial neural network. Eng. Technol. Appl. Sci. Res. 2(1), 167–172 (2012)
Zjavka, L.: Forecast models of partial differential equations using polynomial networks. In: Advances in Intelligent Systems and Computing, Vol. 238, pp. 1–11. Springer, Berlin (2013)
Acknowledgement
This work was supported by the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070), funded by the European Regional Development Fund and the national budget of the Czech Republic via the Research and Development for Innovations Operational Programme and by Project SP2015/146 “Parallel processing of Big data 2” of the Student Grand System, VŠB—Technical University of Ostrava.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Zjavka, L., Snášel, V. (2016). Complex System Modeling with General Differential Equations Solved by Means of Polynomial Networks. In: Stýskala, V., Kolosov, D., Snášel, V., Karakeyev, T., Abraham, A. (eds) Intelligent Systems for Computer Modelling . Advances in Intelligent Systems and Computing, vol 423. Springer, Cham. https://doi.org/10.1007/978-3-319-27644-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-27644-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-27642-7
Online ISBN: 978-3-319-27644-1
eBook Packages: EngineeringEngineering (R0)