SPINI: a structure-preserving neural integrator for hamiltonian dynamics and parametric perturbation

Area of Science:

• Computational Physics
• Numerical Analysis
• Machine Learning

Background:

• Standard numerical solvers fail in long-term simulations of nonlinear Hamiltonian systems.
• These solvers often introduce unphysical errors and lack geometric structure preservation.

Purpose of the Study:

• Introduce a novel symplectic physics-informed neural network integrator (SPINI).
• Develop a robust, law-driven framework for complex computational dynamics.

Main Methods:

• Utilize an unsupervised physics-informed neural network (PINN) to learn the system's Hamiltonian directly from governing equations.
• Embed the learned Hamiltonian surrogate within a 4th-order Yoshida symplectic integrator for structure-preserving time evolution.

Main Results:

• SPINI demonstrates superior accuracy and long-term fidelity compared to analytical solutions and standard solvers like ode45.
• The method excels in strongly nonlinear, large-angle regimes of classical nonlinear pendulums.

Conclusions:

• SPINI offers a robust and accurate approach for simulating nonlinear Hamiltonian systems.
• The hybrid algorithm effectively preserves geometric structure and minimizes unphysical errors in long-term simulations.