In this study, an effective neural network model is proposed using unconstrained optimization based on a newly modified BFGS (Broyden–Fletcher–Goldfarb–Shanno) update algorithm. The fourth-order nonlinear partial differential equation is mathematically formulated through a feed-forward artificial neural network with adaptive parameters. The modification of the BFGS method is introduced to overcome difficulties observed in conventional implementations, which often require significant memory, storage, and computational cost per iteration due to updated Hessian approximations.
The modified BFGS algorithm efficiently estimates the second-order curvature of the goal (energy) function with high precision using gradient and function value data. Theoretical analysis demonstrates improved global convergence properties, where the parameter $p$ in the update formula varies between 0 and 1. Numerical experiments confirm that the modified BFGS update yields higher accuracy and faster convergence compared to the traditional BFGS method. Additionally, a nonmonotone line search technique is used to optimize step length and further enhance performance. The proposed neural network is applied successfully to solve nonlinear PDEs with high reliability and computational efficiency.