Comparing PINN and Symbolic Transform Methods in Modeling the Nonlinear Dynamics of Complex Systems: A Case Study of the Troesch Problem

Nonlinear complex systems exhibit emergent behavior, sensitivity to initial conditions, and rich dynamics arising from interactions among their components. A classical example of such a system is the Troesch problem—a nonlinear boundary value problem with wide applications in physics and engineering. In this work, we investigate and compare two distinct approaches to solving this problem: the Differential Transform Method (DTM), representing an analytical–symbolic technique, and Physics-Informed Neural Networks (PINNs), a neural computation framework inspired by physical system dynamics. The DTM yields a continuous form of the approximate solution, enabling detailed analysis of the system’s dynamics and error control, whereas PINNs, once trained, offer flexible estimation at any point in the domain, embedding the physical model into an adaptive learning process. We evaluate both methods in terms of accuracy, stability, and computational efficiency, with particular focus on their ability to capture key features of nonlinear complex systems. The results demonstrate the potential of combining symbolic and neural approaches in studying emergent dynamics in nonlinear systems.