Dynamic behavior of a dual-spring pendulum: A physics-informed neural network approach

Pendulum systems coupled with springs exhibit nonlinear and coupled dynamics that are difficult to model with purely analytical or numerical approaches. This study investigates a dual-spring pendulum using a hybrid framework that integrates Lagrangian mechanics, Runge–Kutta numerical integration, and physics-informed neural networks (PINNs). The equations of motion are derived via the Lagrangian method, and high-fidelity RK45 solutions are used to pre-train a deep neural network, which is subsequently refined through a physics-based loss function. Comparative simulations under multiple initial conditions demonstrate that the PINN surrogate model preserves phase space structure, achieves high accuracy, and shows strong generalization with errors on the order of 10 −3 . The findings highlight the effectiveness of combining deep learning with classical mechanics for modeling complex oscillatory systems, offering a pathway toward real-time prediction and control of nonlinear mechanical devices.