Feed-Forward Neural Networks Based on the Eigenstates of the Quantum Harmonic Oscillator

Gerasimos Rigatos

doi:10.20965/jaciii.2006.p0567

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JACIII Vol.10 No.4 pp. 567-577

doi: 10.20965/jaciii.2006.p0567

(2006)

Paper:

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Feed-Forward Neural Networks Based on the Eigenstates of the Quantum Harmonic Oscillator

Gerasimos Rigatos

Unit of Industrial Automation, Industrial Systems Institute, 26504, Rion Patras, Greece

Received:

September 1, 2005

Accepted:

January 30, 2006

Published:

July 20, 2006

Keywords:

feed-forward neural networks, quantum harmonic oscillator, Schrödinger’s diffusion equation, Gauss-Hermite expansion, Hermite polynomials

Abstract

The paper introduces feed-forward neural networks where the hidden units employ orthogonal Hermite polynomials for their activation functions. The proposed neural networks have some interesting properties: (i) the basis functions are invariant under the Fourier transform, subject only to a change of scale, (ii) the basis functions are the eigenstates of the quantum harmonic oscillator, and stem from the solution of Schrödinger’s diffusion equation. The proposed feed-forward neural networks belong to the general category of nonparametric estimators and can be used for function approximation, system modelling and image processing.

Cite this article as:

G. Rigatos, “Feed-Forward Neural Networks Based on the Eigenstates of the Quantum Harmonic Oscillator,” J. Adv. Comput. Intell. Intell. Inform., Vol.10 No.4, pp. 567-577, 2006.

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References

[1] C. Cohen-Tannoudji, B. Diu, and F. Laloë, “Mécanique Quantique I,” Hermann, 1998.
[2] W. A. Strauss, “Partial Differential Equations: An Introduction,” J. Wiley, 1992.
[3] B. Kosko, “Neural networks and fuzzy systems: A dynamical systems approach to machine intelligence,” Prentice Hall, 1992.
[4] D. Ventura, and T. Martinez, “Quantum Associative Memory,” Information Sciences, Elsevier, Vol.124, No.1-4, pp. 273-296, 2000.
[5] G. Resconi, and A. J. Van der Waal, “Morphogenic neural networks encode abstract rules by data,” Information Sciences, Elsevier, pp. 249-273, 2002.
[6] M. Perus, “Multi-level Synergetic Computation in Brain,” Nonlinear Phenomena in Complex Systems, Vol.4, No.2, pp. 157-193, 2001.
[7] G. G. Rigatos, and S. G. Tzafestas, “Parallelization of a fuzzy control algorithm using quantum computation,” IEEE Transactions on Fuzzy Systems, Vol.10, No.4, pp. 451-460, 2002.
[8] G. G. Rigatos, and S. G. Tzafestas, “Fuzzy learning compatible with quantum mechanics postulates,” Computational Intelligence and Natural Computation, CINC’03, North Carolina, 2003.
[9] A. Refregier, “Shapelets – I. A method for image analysis,” Mon. Not. R. Astron. Soc., Vol.338, pp. 35-47, 2003.
[10] S. Haykin, “Neural Networks: A Comprehensive Foundation,” McMillan, 1994.
[11] L. Ma, and K. Khorasani, “Constructive Feedforward Neural Networks Using Hermite Polynomial Activation Functions,” IEEE Transactions on Neural Networks, Vol.16, No.4, pp. 821-833, 2005.
[12] L. Ma, and K. Khorasani, “Application of Adaptive Constructive Neural Networks to Image Compression,” IEEE Transactions on Neural Networks, Vol.13, No.5, pp. 1112-1125, 2002.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] C. Cohen-Tannoudji, B. Diu, and F. Laloë, “Mécanique Quantique I,” Hermann, 1998.

[2] [2] W. A. Strauss, “Partial Differential Equations: An Introduction,” J. Wiley, 1992.

[3] [3] B. Kosko, “Neural networks and fuzzy systems: A dynamical systems approach to machine intelligence,” Prentice Hall, 1992.

[4] [4] D. Ventura, and T. Martinez, “Quantum Associative Memory,” Information Sciences, Elsevier, Vol.124, No.1-4, pp. 273-296, 2000.

[5] [5] G. Resconi, and A. J. Van der Waal, “Morphogenic neural networks encode abstract rules by data,” Information Sciences, Elsevier, pp. 249-273, 2002.

[6] [6] M. Perus, “Multi-level Synergetic Computation in Brain,” Nonlinear Phenomena in Complex Systems, Vol.4, No.2, pp. 157-193, 2001.

[7] [7] G. G. Rigatos, and S. G. Tzafestas, “Parallelization of a fuzzy control algorithm using quantum computation,” IEEE Transactions on Fuzzy Systems, Vol.10, No.4, pp. 451-460, 2002.

[8] [8] G. G. Rigatos, and S. G. Tzafestas, “Fuzzy learning compatible with quantum mechanics postulates,” Computational Intelligence and Natural Computation, CINC’03, North Carolina, 2003.

[9] [9] A. Refregier, “Shapelets – I. A method for image analysis,” Mon. Not. R. Astron. Soc., Vol.338, pp. 35-47, 2003.

[10] [10] S. Haykin, “Neural Networks: A Comprehensive Foundation,” McMillan, 1994.

[11] [11] L. Ma, and K. Khorasani, “Constructive Feedforward Neural Networks Using Hermite Polynomial Activation Functions,” IEEE Transactions on Neural Networks, Vol.16, No.4, pp. 821-833, 2005.

[12] [12] L. Ma, and K. Khorasani, “Application of Adaptive Constructive Neural Networks to Image Compression,” IEEE Transactions on Neural Networks, Vol.13, No.5, pp. 1112-1125, 2002.