Modeling And Analysis Of Solving Matrix Equation With Neural Network

Nowadays, the computation speed of the computer is millionfold faster than human brain. It can manage the questions which are well characterized and/or have a clear opera-tional rule with a super-speed. Therefore, the computer greatly expend human brain ability in numerical calculation and logic operation. However, while dealing with the questions un-der complicated environment, e.g. character or image identification, human brain could beat the computer hollow. This is because human brain has a unique structure and billions neu-rons which have great connections and are highly parallel. All of these cover the shortage of the slow computation speed of single neuron. Scientists build a new type of computation system-Artificial Neural Networks(ANN) through imitating human brain.The basic processing unit of ANN is neuron. The first ANN model was put forward in the middle of last century. Up to this time the scientists have build almost a hundred types of ANN models. The connection method and information transfer mode are different between various ANN models. The ANN model used in this paper is one of the most popular models-Hopfield neural network(HNN).In this thesis, we try to solve the Lyapunov matrix equation with the gradient-based neural network(GNN) and to acquire the inverse of a matrix. However, to obtain the inverse of a matrix could also be seen as to solve a matrix equation. People have already constructed the corresponding GNN model and combine it with various types of activation functions to accelerate the convergence for solving the matrix equation. However, all these methods are proven to converge to the accurate solution of the matrix equation only when time goes to infinity. We combine the GNN model with a new type of activation function named by Li activation function. The global convergence and finite-time convergence are proven in theory. The upper bound of the convergence time is also given analytically. In the end, numerical simulations are given to validate the theory conclusion.