SPINI: A Structure-Preserving Neural Integrator for Hamiltonian Dynamics and Parametric Resonance

Standard numerical solvers struggle with the long-term simulation of nonlinear Hamiltonian systems, often failing to preserve geometric structure and introducing unphysical errors. This paper introduces the Symplectic Physics-Informed Neural Network Integrator (SPINI), a novel two-stage hybrid algorithm. First, an unsupervised Physics-Informed Neural Network (PINN) learns the system's Hamiltonian directly from its governing equations, requiring no trajectory data. Second, this learned Hamiltonian surrogate is embedded within a 4th-order Yoshida symplectic integrator to ensure a structure-preserving time evolution. We apply SPINI to the classical nonlinear pendulum and its parametric resonance. Validations against the analytical solution and a standard Runge-Kutta solver (ode45) demonstrate SPINI's superior accuracy and long-term fidelity, particularly in the strongly nonlinear, large-angle regime. SPINI offers a robust, law-driven framework for complex computational dynamics.