Abstract
A general topological approach is proposed for the construction of converging artificial neural networks (ANN) by applying decision-making algorithms tuned on a sequence of iterations of continuous mappings (ANN layers). The mappings are selected using optimization principles underlying ANN training, and decision-making based on the results of training a multilayer ANN corresponds to finding a sequence converging to a fixed point. It is found that problems of this class are computationally unstable, which is caused by the phenomenon of dynamic chaos associated with the ill-posedness of the problems. Stabilization methods converging to stable fixed points of the mappings are proposed, which is the starting point for a wide variety of mathematical studies concerning the optimization of training sets in ANN construction.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
V. B. Betelin and V. A. Galkin, “Mathematical problems related to artificial intelligence and artificial neural networks,” Usp. Kibern. 2 (4), 6–14 (2021).
A. V. Kryanev, G. V. Lukin, and D. K. Udumyan, Metric Analysis and Data Processing (Fizmatgiz, Moscow, 2012) [in Russian].
T. Y. Ly and J. A. Yorke, “Period three implies chaos,” Am. Math. Mon. 82, 982–985 (1975).
D. V. Anosov, “Geodesic flows on closed Riemannian manifolds of negative curvature,” Proc. Steklov Inst. Math. 90, 1–235 (1967).
A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Dover, New York, 1999).
N. Dunford and J. T. Schwartz, Linear Operators, Vol. 1: General Theory (Interscience, New York, 1958).
E. S. Nikolaev and A. A. Samarskii, “Selection of the iterative parameters in Richardson’s method,” USSR Comput. Math. Math. Phys. 12 (4), 141–158 (1972).
P. S. Aleksandrov and B. A. Pasynkov, Introduction to Dimension Theory (Nauka, Moscow, 1973) [in Russian].
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge Univ. Press, Cambridge, 1995).
Funding
This work was performed within the state assignment of the Scientific Research Institute for System Analysis of the Russian Academy of Sciences (fundamental research GP-47), subject no. 0065-2019-0007.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare that they have no conflicts of interest.
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Betelin, V.B., Galkin, V.A. On Fixed Points of Continuous Mappings Associated with Construction of Artificial Neural Networks. Dokl. Math. 106, 423–425 (2022). https://doi.org/10.1134/S1064562422700089
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562422700089