Comparative analysis of the RK4 and hybrid RK4-feedforward neural network method for solving second-order ordinary differential equation initial value problems

This study presents a hybrid numerical method that combines the classical fourth-order Runge–Kutta (RK4) method with a feedforward neural network (FNN) to solve second-order initial value problems of ordinary differential equations. The FNN learns the numerical error of the RK4 solution and provides a weighted correction controlled by the parameter λ. The proposed method was tested on seven benchmark problems covering linear, nonlinear, stiff, oscillatory, Euler–Cauchy, chaotic, and coupled systems. Its performance was evaluated in terms of numerical accuracy, convergence, stability, statistical reliability, and computational efficiency. The results showed that the RK4–FNN method generally reduced numerical errors, particularly for coarse step sizes and complex problems. Decreasing the step size improved the accuracy of both methods and reduced the influence of the neural correction. Statistical analysis indicated that moderate correction values provided lower variability and more reliable results than excessive corrections. The optimal λ values were selected using a cost function that balances accuracy and statistical stability. Although the hybrid method required additional computation, its memory overhead remained low.