Novel Neural Network Based on New Modification of BFGS Update Algorithm for Solving Partial Differential Equations

Abstract

In this article, an effective neural network is created using unconstrained optimization the brand-new BFGS training algorithm. The fourth order nonlinear partial differential equation is mathematically modeled with feed-forward artificial neural network with some adaptive parameters. The network is trained by new modification of BFGS method to avoid some troubles occurs when the network trained by current BFGS. The conventional updated Hessian approximations approach needed significant memory, storage, and cost computing for each iteration. One of these update’s novel features is its ability to estimate the 2nd order curvature of the goal function (energy functions) with high order precision while using the provided gradient and function value data. It is shown that the global convergence properties of the suggested modification, there is a parameter ρ in the update formulae which ranges from zero to one. The numerical experiments demonstrate that the improved BFGS update will be more accurate and more effective than the traditional BFGS methods. The proposed algorithm has well properties such: it has global convergence for energy function which is convex functions; also to get optimal step length we used a nonmonotone line search technique to modify the effectiveness of the proposed algorithm. Finally, used suggested training algorithm, to learned an appropriate neural network for accurately solving any non-linear PDEs.