Feedforward Neural Network with BFGS Optimization for Solving Ordinary Differential Equations with Application to Tumor-Immune Dynamics

Abstract

Ordinary Differential Equations (ODEs) are fundamental for modeling dynamic systems in science and engineering. Traditional numerical methods, such as Euler and Runge-Kutta, are widely used but may face limitations in accuracy and computational efficiency, particularly for nonlinear problems. Recently, one of the branches in Neural Network known as feedforward neural network (FNN) have emerged as an alternative approach for solving differential equations due to their universal approximation capability. This study proposes a framework for solving ODEs by integrating a FNN with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization algorithm. The method constructs a trial solution that automatically satisfies the initial conditions and trains the network by minimizing a loss function defined by the residual of the differential equation. The proposed method is tested on several linear and nonlinear ODE problems and is further applied to a tumor-immune-drug interaction model to illustrate its capability in handling biologically motivated dynamical systems. Numerical results demonstrate that the ANN-BFGS approach achieves higher accuracy than Euler and fourth-order Runge-Kutta (RK4) methods while providing a continuous and differentiable solution across the domain.