Markov chain network training and conservation law approximations: Linking microscopic and macroscopic models for evolution

Applied Mathematics and Computation

Volumn 199, Issue 1, 2008, Pages 315-333

  Melnik, Roderick V N ^a  

^a WILFRID LAURIER UNIVERSITY   (Canada)

Author keywords

Complex dynamic systems; Conservation law approximations; Markov chain approximation method; Neural networks; Training dynamics

Indexed keywords

APPROXIMATION THEORY; DYNAMICAL SYSTEMS; MARKOV PROCESSES;

CONSERVATION LAW APPROXIMATIONS; MARKOV CHAIN APPROXIMATION METHOD;

NEURAL NETWORKS;

EID: 41949115398     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2007.09.063     Document Type: Article

References (37)

1. Agogino A., Stanley K., and Mikkulainen R. Online interactive neuro-evolution. Neural Process. Lett. 11 (2000) 29-37
2. Aguilar J. Learning algorithm retrieval process for the multiple classes random neural network model. Neural Process. Lett. 13 (2001) 81-91
3. Ahmed M.S., and Riyaz S.H. Dynamic observers - a neural net approach. J. Intell. Fuzzy Syst. 9 (2000) 113-127
4. Balakrishnan S.N., and Biega V. Adaptive-critic-based neural networks for aircraft optimal control. J. Guid. Control Dynam. 19 (1996) 893-989
5. Barron A.R. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory 39 (1993) 930-945
6. Bishop C. Neural Networks for Pattern Recognition (1995), Clarendon Press, Oxford
7. Cabrera J.B.D., and Narendra K.S. Issues in the application of neural networks for tracking based on inverse control. IEEE Trans. Autom. Contr. 44 (1999) 2007-2027
8. Catala A., and Angulo C. A comparison between the Tikhonov and the Bayesian approaches to calculate regularisation matrices. Neural Process. Lett. 11 (2000) 185-195
9. Chaitin G.J. Godel's theorem and information. Int. J. Theor. Phys. 22 (1982) 941-954
10. Corinto F., Biey M., and Gilli M. Non-linear coupled CNN models for multiscale image analysis. Int. J. Circuit Theory Appl. 34 1 (2006) 77-88
11. Costa P., and Larzabal P. Initialization of supervised training for parametric estimation. Neural Process. Lett. 9 (1999) 53-61
12. Croft H.T., Falconer K.J., and Guy R.K. Unsolved Problems in Geometry (1991), Springer-Verlag, New York p. 3
13. Cybenko G. Approximation by superposition of sigmoidal functions. MCSS 2 (1989) 303-314
14. DasGupta B., and Hammer B. On approximate learning by multi-layered feedforward circuits. Theor. Comput. Sci. 348 1 (2005) 95-127
15. Efe M.O., and Kaynak O. Stabilizing and robustifying the learning mechanisms of ANNs in control engineering applications. Int. J. Intell. Syst. 15 (2000) 365-388
16. Gawin D., Lefik M., and Schrefler B.A. ANN approach to sorption hysteresis within a coupled hydro-thermo-mechanical FE analysis. Int. J. Numer. Method Eng. 50 (2001) 299-323
17. Guenther R.B., and Lee J.W. Partial Differential Equations of Mathematical Physics and Integral Equations (1988), Prentice Hall, NJ
18. Hornik K., Stinchcombe M., and White H. Multi-layer feedforward networks are universal approximators. Neural Network 2 (1989) 359-366
19. Lapidus L., and Pinder G.F. Numerical Solution of PDEs in Science and Engineering (1999), John Wiley & Sons
20. Levin A.U., and Narendra K.S. Control of nonlinear dynamical systems using neural networks. Part I. IEEE Trans. Neural Network 4 (1993) 192-206
21. Levin A.U., and Narendra K.S. Control of nonlinear dynamical systems using neural networks. Part II. IEEE Trans. Neural Network 7 (1996) 30-42
22. Liao L.-Z. A recurrent neural network for N-stage optimal control problems. Neural Process. Lett. 10 (1999) 195-200
23. R.V.N. Melnik, A hierarchy of hyperbolic macrodynamic equations as a model for network training, in: Proceedings of the IEEE International Symposium on Information Theory, Ulm, Germany, June 29-July 4, 1997, p. 322.
24. Melnik R.V.N. On consistent regularities of control and value functions. Numer. Funct. Anal. Optim. 18 (1997) 401-426
25. Melnik R.V.N. Dynamic system evolution and Markov chain approximation. Discr. Dynam. Nat. Soc. 2 (1998) 7-39
26. Melnik R.V.N. Deterministic and stochastic dynamics with hyperbolic HJB-type equations. Dynam. Contin. Discr. Impul. Syst. Ser. A: Math. Anal. 10 3 (2003) 317-330
27. Müller I., and Ruggeri T. Extended Thermodynamics (1993), Springer-Verlag
28. Narendra K.S. Neural networks for control: theory and practice. Proc. IEEE 84 (1996) 1385-1406
29. Narendra K.S., and Parthasarathy K. Identification and control of dynamical systems using neural networks. IEEE Trans. Neural Network 1 (1990) 4-27
30. Narendra K.S., and Xiang C. Adaptive control of discrete-time systems using multiple models. IEEE Trans. Autom. Contr. 45 (2000) 1669-1686
31. Quarteroni A., Sacco R., and Saleri F. Numerical Mathematics (2000), Springer
32. Santini S., and Del Bimbo A. Properties of block feedback neural networks. Neural Network 8 4 (1995) 579-596
33. Sendhoff B., and Kreutz M. A model for the dynamic interaction between evolution and learning. Neural Process. Lett. 10 (1999) 181-193
34. Ulmer II C.W., et al. Computational neural network and the rational design of polymeric materials. Comput. Theor. Polym. Sci. 8 (1998) 311-321
35. Wahba G. Generalization and regularization of nonlinear learning systems. Handbook of Brain Theory and Neural Networks. second ed. (2000), M. Arib
36. Zhang J., et al. Prediction of polymer quality in batch polymerisation reactions using robust neural networks. Chem. Eng. J. 69 (1998) 135-143
37. Zhang T., Ge S.S., and Hang C.C. Neural-based direct adaptive control for a class of general nonlinear systems. Int. J. Syst. Sci. 28 (1997) 1011-1020