Abstract
A numerical method for computational implementation of gradient dynamical systems is presented. The method is based upon the development of geometric integration numerical methods, which aim at preserving the dynamical properties of the original ordinary differential equation under discretization. In particular, the proposed method belongs to the class of discrete gradients methods, which substitute the gradient of the continuous equation with a discrete gradient, leading to a map that possesses the same Lyapunov function of the dynamical system, thus preserving the qualitative properties regardless of the step size. In this work, we apply a discrete gradient method to the implementation of Hopfield neural networks. Contrary to most geometric integration methods, the proposed algorithm can be rewritten in explicit form, which considerably improves its performance and stability. Simulation results show that the preservation of the Lyapunov function leads to an improved performance, compared to the conventional discretization.
This work has been partially supported by FEDER funds (through project no. TIN2010-16556 from the Spanish Government and project no. P08-TIC-04026 from the Junta de Andalucía), and the Agencia Española de Cooperación Internacional para el Desarrollo (project no. A2/038418/11).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abe, S.: Theories on the Hopfield Neural Networks. In: IEE International Joint Conference on Neural Networks, vol. I, pp. 557–564 (1989)
Atencia, M.A., Joya, G., Sandoval, F.: Dynamical Analysis of Continuous Higher Order Hopfield Networks for Combinatorial Optimization. Neural Computation 17(8), 1802–1819 (2005)
Atencia, M.A., Joya, G., Sandoval, F.: Hopfield Neural Networks for Parametric Identification of Dynamical Systems. Neural Processing Letters 21(2), 143–152 (2005)
van den Bos, A.: Parameter estimation for scientists and engineers. Wiley-Interscience (2007)
Calvo, M., Laburta, M.P., Montijano, J.I., Rández, L.: Projection methods preserving lyapunov functions. BIT Numerical Mathematics 50(2), 223–241 (2010)
Hairer, E.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2006)
Haykin, S.: Neural Networks. A Comprehensive Foundation. Macmillan College Publishing Company (1994)
Hopfield, J.: Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982)
Hopfield, J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984)
Iserles, A., Peplow, A., Stuart, A.: A unified approach to spurious solutions introduced by time discretisation. Part I: Basic theory. SIAM J. Numer. Anal. 28(6), 1723–1751 (1991)
Khalil, H.K.: Nonlinear Systems. Prentice Hall (2002)
McLachlan, R., Quispel, R., Robidoux, N.: Geometric integration using discrete gradients. Philos. Trans. of the Royal Society of London Series A 357(1754), 1021–1045 (1999)
Stuart, A., Humphries, A.: Dynamical systems and numerical analysis. Cambridge University Press (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Atencia, M., Hernández, Y., Joya, G., Sandoval, F. (2013). Numerical Implementation of Gradient Algorithms. In: Rojas, I., Joya, G., Cabestany, J. (eds) Advances in Computational Intelligence. IWANN 2013. Lecture Notes in Computer Science, vol 7903. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38682-4_38
Download citation
DOI: https://doi.org/10.1007/978-3-642-38682-4_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38681-7
Online ISBN: 978-3-642-38682-4
eBook Packages: Computer ScienceComputer Science (R0)