Application of high-order hopfield neural networks to the solution of diophantine equations
Hopfield and Tank network with high-order weights is applied to the solution of algebraic problems. Particularly, to the search of positive integer solution of a diophantine equation. The chosen representation avoids using all the possible connections among neurons, so reducing one of the most serious problems of high order: combinatorial growing of connections. The energy function is found to be polynomial of order 2n-1 where n is the order of the equation. Although each network is problem-specific, the building process may be extended to other similar problems without any difficulty.
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- .-H. H. Chen et al. "High-order correlation using a model for associative memory". In AIP Conf. Proc. 151, Ed. J.S. Denker, p.86, American Institute of Physics (1986).Google Scholar
- .-M. R. Garey and D. S. Johnson. "Computers and intractability. A Guide to the theory of NP-Completeness". W.H.Freeman and Company (1979)Google Scholar
- .-C. L. Giles and T. Maxwell. "Learning, invariance, and generalization in high-order neural networks". Applied Optics, Vol. 26, No. 23, pp. 4972–4978, (1987)Google Scholar
- .-J. Hertz et al. "Introduction to the theory of neural computation". Addison-Wesley (1991)Google Scholar
- .-Heng-Ming Tai et al. "Information Storage in High-order neural networks using unequal neural activity". Journal of the Franklin Institute, Vol. 327, No. 1, pp. 129–141, (1990)Google Scholar
- .-J. J. Hopfield. "Neural networks and physical systems with emergent collective computational abilities" Proc. Nat. Ac. Sc. USA, Vol. 79, pp. 2554–2558 (1982)Google Scholar
- .-J. J. Hopfield. "Neurons with graded response have collective computational properties like those of two state neurons". Proc. Nat. Ac. Sc. USA, Vol. 81, pp. 3088–3092, (1984)Google Scholar
- .-J. J. Hopfield and D. W. Tank. "Neural computation of decisions in optimization problems". Biological Cybernetics, Vol. 52, pp. 141–152, (1985)Google Scholar
- .-R. P. Lippman. "An introduction to computing with neural nets". IEEE ASSP Magazine, pp. 4–22, April (1987)Google Scholar
- .-T. Samad and P. Harper. "High-order Hopfield and Tank optimization networks". Parallel Computing, Vol. 16, pp, 287–292, (1990)Google Scholar
- .-T. J. Sejnowsky. "High-order Boltzmann machines". AIP Conf. Proc. 151, pp.398–403, Ed. J.S. Denker, American Institute of Physics (1986)Google Scholar
- .-N.H. Goddard et Al, "Rochester Connectionist Simulator", Technical Report 233, The University of Rochester (USA), (1989)Google Scholar