Three-dimensional structured networks for matrix equation solving

Structured networks are feedforward neural networks with linear neurons than use special training algorithms. Two three-dimensional (3-D) structured networks are developed for solving linear equations and the Lyapunov equation. The basic idea of the structured network approaches is: first, represent a given equation-solving problem by a 3-D structured network so that if the network matches a desired pattern array, the weights of the linear neurons give the solution to the problem; then, train the 3-D structured network to match the desired pattern array using some training algorithms; finally, obtain the solution to the specific problem from the converged weights of the network. The training algorithms for the two 3-D structured networks are proved to converge exponentially fast to the correct solutions