Solving Nonlinear Equations Using Recurrent Neural Networks

A class of recurrent neural networks is developed to solve nonlinear equations, which are approximated by a multilayer perceptron (MLP). The recurrent network includes a linear Hopfield network (LHN) and the MLP as building blocks. This network inverts the original MLP using constrained linear optimization and Newton’s method for nonlinear systems. The solution of a nonlinear equation with computer simulation illustrates the algorithm. 1 Problem Context A class of recurrent neural networks is used to solve nonlinear equations, where the motivation for using artificial neural networks is their learning capability. For this work the properties of multilayer perceptron (MLP) as universal function approximators [3] are of particular advantage: data with unknown structure, taken from a nonlinear system, are approximated by an MLP. The MLP is then inverted, i.e. the approximated equation is solved, using recurrent neural networks. This class of recurrent networks is comprised of MLPs and LHN [4] as building blocks. Finding the solution of n-dimensional nonlinear equations can be a challenging problem, even when a unique solution exists. The recurrent networks presented here employ Newton’s method for nonlinear systems [1] for the solution of such systems. The LHN perform linear optimization, and if necessary, a generalized linear Hopfield network is used to perform constrained linear optimization [5]. The solution of a nonlinear equation is presented as an application example.