Abstract
The main contribution of this paper is using optimal control theory for improving the convergence rate of backpropagation algorithm. In the proposed approach, the learning algorithm of backpropagation is modeled as a minimum time control problem in which the step-size of its learning factor is considered as the input of this model. In contrast to the traditional backpropagation, learning algorithms which select the step-size by trial and error, it is selected adaptively based on optimal control criterion. The effectiveness of the proposed algorithm is evaluated in two simulations: XOR and 3-bit parity. In both simulation examples, the proposed algorithm outperforms well in speed and the ability to escape from local minima.
Similar content being viewed by others
References
Ahn, C.K.: L 2–L ∞ nonlinear system identification via recurrent neural networks. Nonlinear Dyn. 62(3), 543–552 (2010)
Ahn, C.K.: An H∞ approach to stability analysis of switched Hopfield neural networks with time-delay. Nonlinear Dyn. 60(4), 703–711 (2010)
Alves, E.: Earthquake forecasting using neural networks: results and future work. Nonlinear Dyn. 44(1), 341–349 (2006)
Roopaei, M., Zolghadri Jahromi, M., Ranjbar-Sahraei, B., Lin, T.C.: Synchronization of two different chaotic systems using novel adaptive interval type-2 fuzzy sliding mode control. Nonlinear Dyn. 66(4), 667–680 (2011)
Rumelhart, D.E., McClelland, J.L.: Parallel Distributed Processing: Exploration in the Microstructure of Cognition, vol. 1. MIT Press, Cambridge (1986)
Hertz, J., Krough, A., Palmer, R.G.: Introduction to the Theory of Neural Computation. Addison-Wesley, Reading (1991)
Baba, N., Handa, H.: Utilization of hierarchical structure stochastic automata for the back propagation method with momentum. In: IEEE International Conference on Neural Networks, vol. 1, pp. 389–393 (1995)
Zweiri, Y.H., Whidborn, J.F., Senevirstne, L.D.: A three-term backpropagation algorithm. Neurocomputing 50, 305–318 (2003)
Bhaya, A., Kaszkurewicz, E.: A control-theoretic approach to the design of zero finding numerical methods. IEEE Trans. Autom. Control 52(6), 1014–1026 (2007)
Zweiri, Y.H., Seneviratne, L.D., Althoefer, K.: Stability analysis of a three-term backpropagation algorithm. Neural Netw. 18, 1341–1347 (2005)
Behera, L., Kumar, S., Patnaik, A.: On adaptive learning rate that guarantees convergence in feedforward networks. IEEE Trans. Neural Netw. 17(5) (2006)
Man, Z., Wu, H.R., Liu, S., Yu, X.: A new adaptive backpropagation algorithm based on Lyapunov stability theory for neural network. IEEE Trans. Neural Netw. 17(6) (2006)
Jacobs, R.A.: Increasing rate of convergence through learning rate adaptation. Neural Netw. 1, 295–307 (1981)
Sira-Ramirezand, H., Colina-morlez, E.: A sliding mode strategy for adaptive learning in Adalines. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 42(12), 1001–1012 (1995)
Poznyak, E.N., Sanchez, E.N., Yu, W.: Differential Neural Networks for Robust Nonlinear Control-Identification, State Estimation and Trajectory Tracking. Word Scientific, Singapore (2001)
Lewis, F.L., Syrmos, V.L.: Optimal Control. Wiley, New York (1995). ISBN:0471033782
Kirk, D.E.: Optimal Control Theory. Dover, New York (2004). ISBN:0486434842
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jahangir, M., Golshan, M., Khosravi, S. et al. Design of a fast convergent backpropagation algorithm based on optimal control theory. Nonlinear Dyn 70, 1051–1059 (2012). https://doi.org/10.1007/s11071-012-0512-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-012-0512-1