Qualitative behavior of differential equations associated with artificial neural networks
Here, the following matrix differential equation is considered \[ \frac{dW}{dt}=T_{\varepsilon}CW-WW^TCW, \tag{1} \] where \(W(t)\) is a real \(n\times k\)-matrix, \(C\) is a symmetric matrix and \(T_{\varepsilon}\) is a tridiagonal matrix. Equation (1) is proposed by Oja, Kingsley and Adams as a learning model, more precisely, a model that describes the dynamics of an artificial neural network. The elements of the matrix \(W\) are related with the change of information between the neurons, while the parameter \(\varepsilon\) describes the probability of temporary synaptic formation. Using techniques from matrix calculus, the authors establish conditions under which (1) is a gradient, semi-gradient or a gradient-like system. They also study the asymptotic behavior of solutions of system (1).
scientific article; zbMATH DE number 2128705