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Nonlinear dynamical system modeling via recurrent neural networks and a weighted state space search algorithm
1.  Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China 
2.  Department of Mathematics, Cleveland State University, Cleveland, OH 44115, United States 
References:
[1] 
A. F. Atiya and A. G. Parlos, New results on recurrent network training: Unifying the algorithms and accelerating convergence, IEEE Transcations on Neural Networks, 11 (2000), 697709. doi: 10.1109/72.846741. 
[2] 
Y. Fang and T. W. S. Chow, Nonlinear dynamical systems control using a new RNN temporal learning strategy, IEEE Trans on Circuit and Systems, Part II, 52 (2005), 719723. 
[3] 
R. A. Conn, K. Scheinberg and N. L. Vicente, "Introduction to Derivativefree Optimization," SIAM, 2009. doi: 10.1137/1.9780898718768. 
[4] 
J. F. G. Freitas, M. Niranjan, A. H. Gee and A. Doucet, Sequential Monte Carlo methods to train neural network models, Neural Computation, 12 (2000), 955993. doi: 10.1162/089976600300015664. 
[5] 
L. K. Li, Learning sunspot series dynamics by recurrent neural networks, in "Advances in Data Mining and Modeling" (eds. W. K. Ching and K. P. Ng), World Science, (2003), 107115. doi: 10.1142/9789812704955_0009. 
[6] 
L. K. Li, W. K. Pang, W. T. Yu and M. D. Trout, Forecasting shortterm exchange Rates: a recurrent neural network approach, in "Neural Networks in Business Forecasting" (eds. G. P. Zhang), Idea Group Publishing, (2004), 195212. doi: 10.4018/9781591401766.ch010. 
[7] 
L. K. Li and S. Shao, Dynamic properties of recurrent neural networks and its approximations, International Journal of Pure and Applied Mathematics, 39 (2007), 545562. 
[8] 
L. K. Li and S. Shao, A neural network approach for global optimization with applications, Neural Network World, 18 (2008), 365379. 
[9] 
L. K. Li, S. Shao and T. Zheleva, A state space search algorithm and its application to learn the shortterm foreign exchange rates, Applied Mathematical Sciences, 2 (2008), 17051728. 
[10] 
X. D. Li, J. K. L. Ho and T. W. S. Chow, Approximation of dynamical timevariant systems by continuoustime recurrent neural networks, IEEE Trans on Circuit and Systems, Part II, 52 (2005), 656660. 
[11] 
X. B. Liang and J. Wang, A recurrent neural network for nonlinear optimization with a continuously differentiable objective function and bound constraints, IEEE Transactions on Neural Networks, 11 (2000), 12511262. doi: 10.1109/72.883412. 
[12] 
Z. Liu and I. Elhanany, A Fast and Scalable Recurrent Neural Network Based on Stochastic Meta Descent, IEEE Transactions on Neural Networks, 19 (2008), 16521658. doi: 10.1109/TNN.2008.2000838. 
[13] 
S. Wang, Q. Shao and X. Zhou, Knotoptimizing spline networks (KOSNETS) for nonparametric regression, Journal of Industrial and Management Optimization, 4 (2008), 33?52. 
[14] 
X. Wang and E. K. Blum, Discretetime versus continuoustime models of neural networks, Journal of Computer and System Sciences, 45 (1992), 119. doi: 10.1016/00220000(92)90038K. 
[15] 
R. J. Williams and D. Zipser, A learning algorithm for continually running fully recurrent neural networks, Neural Computation, 1 (1989), 270280. doi: 10.1162/neco.1989.1.2.270. 
[16] 
L. Xu and W. Liu, A new recurrent neural network adaptive approach for hostgate way rate control protocol within intranets using ATM ABR service, Journal of Industrial and Management Optimization, 1 (2005), 389404. 
[17] 
J. Yao and C. L. Tan, A case study on using neural networks to perform technical forecasting of forex, Neural Computation, 34 (2000), 7998. 
[18] 
K. F. C. Yiu, S. Wang, K. L. Teo and A. H. Tsoi, Nonlinear system modeling via knotoptimizing Bspline networks, IEEE Transactions on Neural Networks, 12 (2001), 10131022. doi: 10.1109/72.950131. 
[19] 
K. F. C. Yiu, Y. Liu and K. L. Teo, A hybrid descent method for global optimization, Journal of Global Optimization, 28 (2004), 229238. doi: 10.1023/B:JOGO.0000015313.93974.b0. 
show all references
References:
[1] 
A. F. Atiya and A. G. Parlos, New results on recurrent network training: Unifying the algorithms and accelerating convergence, IEEE Transcations on Neural Networks, 11 (2000), 697709. doi: 10.1109/72.846741. 
[2] 
Y. Fang and T. W. S. Chow, Nonlinear dynamical systems control using a new RNN temporal learning strategy, IEEE Trans on Circuit and Systems, Part II, 52 (2005), 719723. 
[3] 
R. A. Conn, K. Scheinberg and N. L. Vicente, "Introduction to Derivativefree Optimization," SIAM, 2009. doi: 10.1137/1.9780898718768. 
[4] 
J. F. G. Freitas, M. Niranjan, A. H. Gee and A. Doucet, Sequential Monte Carlo methods to train neural network models, Neural Computation, 12 (2000), 955993. doi: 10.1162/089976600300015664. 
[5] 
L. K. Li, Learning sunspot series dynamics by recurrent neural networks, in "Advances in Data Mining and Modeling" (eds. W. K. Ching and K. P. Ng), World Science, (2003), 107115. doi: 10.1142/9789812704955_0009. 
[6] 
L. K. Li, W. K. Pang, W. T. Yu and M. D. Trout, Forecasting shortterm exchange Rates: a recurrent neural network approach, in "Neural Networks in Business Forecasting" (eds. G. P. Zhang), Idea Group Publishing, (2004), 195212. doi: 10.4018/9781591401766.ch010. 
[7] 
L. K. Li and S. Shao, Dynamic properties of recurrent neural networks and its approximations, International Journal of Pure and Applied Mathematics, 39 (2007), 545562. 
[8] 
L. K. Li and S. Shao, A neural network approach for global optimization with applications, Neural Network World, 18 (2008), 365379. 
[9] 
L. K. Li, S. Shao and T. Zheleva, A state space search algorithm and its application to learn the shortterm foreign exchange rates, Applied Mathematical Sciences, 2 (2008), 17051728. 
[10] 
X. D. Li, J. K. L. Ho and T. W. S. Chow, Approximation of dynamical timevariant systems by continuoustime recurrent neural networks, IEEE Trans on Circuit and Systems, Part II, 52 (2005), 656660. 
[11] 
X. B. Liang and J. Wang, A recurrent neural network for nonlinear optimization with a continuously differentiable objective function and bound constraints, IEEE Transactions on Neural Networks, 11 (2000), 12511262. doi: 10.1109/72.883412. 
[12] 
Z. Liu and I. Elhanany, A Fast and Scalable Recurrent Neural Network Based on Stochastic Meta Descent, IEEE Transactions on Neural Networks, 19 (2008), 16521658. doi: 10.1109/TNN.2008.2000838. 
[13] 
S. Wang, Q. Shao and X. Zhou, Knotoptimizing spline networks (KOSNETS) for nonparametric regression, Journal of Industrial and Management Optimization, 4 (2008), 33?52. 
[14] 
X. Wang and E. K. Blum, Discretetime versus continuoustime models of neural networks, Journal of Computer and System Sciences, 45 (1992), 119. doi: 10.1016/00220000(92)90038K. 
[15] 
R. J. Williams and D. Zipser, A learning algorithm for continually running fully recurrent neural networks, Neural Computation, 1 (1989), 270280. doi: 10.1162/neco.1989.1.2.270. 
[16] 
L. Xu and W. Liu, A new recurrent neural network adaptive approach for hostgate way rate control protocol within intranets using ATM ABR service, Journal of Industrial and Management Optimization, 1 (2005), 389404. 
[17] 
J. Yao and C. L. Tan, A case study on using neural networks to perform technical forecasting of forex, Neural Computation, 34 (2000), 7998. 
[18] 
K. F. C. Yiu, S. Wang, K. L. Teo and A. H. Tsoi, Nonlinear system modeling via knotoptimizing Bspline networks, IEEE Transactions on Neural Networks, 12 (2001), 10131022. doi: 10.1109/72.950131. 
[19] 
K. F. C. Yiu, Y. Liu and K. L. Teo, A hybrid descent method for global optimization, Journal of Global Optimization, 28 (2004), 229238. doi: 10.1023/B:JOGO.0000015313.93974.b0. 
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