• PINN reduces long-term error by over 90% compared to Ms-DTM • Absolute errors remain below 10⁻³ for all nonlinear test cases • Stable predictions achieved over 20 s simulation horizon • Energy drift maintained below 10⁻⁴ in conservative dynamics • Training time limited to ∼13 s for high-accuracy PINN models This study investigates the dynamic behavior of two nonlinearly coupled pendulums using three complementary approaches: the numerical Runge–Kutta (RK45) method, the analytical Multi-Stage Differential Transform Method (Ms-DTM), and the Physics-Informed Neural Network (PINN). The PINN framework integrates fundamental physical laws into its learning process, enabling it to accurately capture the system’s nonlinear oscillations and predict its motion over extended time intervals. The comparative analysis reveals that all three methods produce consistent results, while the PINN exhibits superior long-term stability and robustness in representing highly nonlinear dynamics with RMSE values on the order of 10 −5 for all state variables α 1 , α ˙ 1 , α 2 , α ˙ 2 , physics residual norms of 10 −3 , maximum energy drift of order 10 −4 relative to the RK45 reference. Phase portraits further confirm that the coupled pendulum system conserves energy and remains conservative. These findings demonstrate the effectiveness of hybrid modeling, which combines data-driven learning with physical constraints, providing a powerful tool for analyzing complex nonlinear mechanical systems where traditional analytical solutions are challenging. Moreover, the proposed framework can be extended to other nonlinear oscillatory systems, paving the way for improved modeling accuracy in future studies of dynamic and energy-transfer phenomena.
Eid et al. (Sun,) studied this question.