Approximate Methods for Solving Problems of Mathematical Physics on Neural Hopfield Networks

Ilya Boykov, Vladimir Roudnev and Alla Boykova
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Ilya Boykov: Department of Higher and Applied Mathematics, Penza State University, 40 Krasnaya Str., 440026 Penza, Russia
Vladimir Roudnev: Department of Computational Physics, Saint Petersburg State University, 1 Ulyanovskaya Str., 198504 Saint Petersburg, Russia
Alla Boykova: Department of Higher and Applied Mathematics, Penza State University, 40 Krasnaya Str., 440026 Penza, Russia

Mathematics, 2022, vol. 10, issue 13, 1-22

Abstract: A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular integral equations, using a continuous method for solving operator equations. This method assumes that the original system is associated with a Cauchy problem for systems of ordinary differential equations on Hopfield neural networks. We present sufficient conditions for the Hopfield networks’ stability defined via coefficients of systems of differential equations.

Keywords: Hopfield neural network; singular; hypersingular integral equations; nonlinear differential equations; stability; Cauchy problem; continuous method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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