Backpropagation and Levenberg-Marquardt Algorithm for Training Finite Element Neural Network

In this paper, finite element based neural network is developed. The purpose is to solve differential equation and inverse problem of differential equation. Inverse problem of differential equation is a problem to solve for parameters of differential equation, assuming that the solution of the differential equation is already known beforehand. Inverse problem mainly used to approximate physical parameters of material. Finite element method will be combined with artificial neural network using back propagation algorithm to solve differential equation and Levenberg-Marquardt training algorithm to solve inverse differential problem. By using proposed method, inverse matrix calculation will not be needed for solving both differential equation and inverse differential problem. From any given differential equation, the solution will be solved first. And the solution is used to validate the parameter in differential equation, namely to solve inverse problem of that differential equation.