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

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.

The Effect of Blue Noise on the Optimization Ability of Hopfield Neural Network

“…Optimization problems are divided into two kinds of function optimization and combinatorial optimization. Many practical problems can be categorized into one of these for solving [37,38]. To verify the immunity of Hopfield neural network to colored noise, blue noise was selected to perturb the input and output, respectively, and the effect of noise on Hopfield function optimization and combinatorial optimization problems was explored.…”

“…Let the simulation program take the 30 cities after classical normalization. Let the emulator take the classic 30 cities [48], and the coordinates are as follows: (41,94); (37,84) (45,21); (41,26); (44,35); and (4,50). The shortest path of the 30-city TSP problem is 423.7406, and its shortest path is shown in Figure 8.…”