Symplectic Gaussian process regression of maps in Hamiltonian systems.

Area of Science:

• * Computational Physics
• * Applied Mathematics
• * Machine Learning

Background:

• * Hamiltonian dynamics govern many physical systems, including particle accelerators and plasma confinement.
• * Accurate long-term prediction of these systems is computationally challenging.
• * Existing methods often struggle with stability and accuracy over extended periods.

Purpose of the Study:

• * To develop structure-preserving emulators for Hamiltonian and Poincaré maps.
• * To enable accurate long-term tracing of charged particles and magnetic field lines.
• * To learn system dynamics directly from observational data.

Main Methods:

• * Multi-output Gaussian Process (GP) regression on scattered orbit data.
• * Enforcing symplectic properties via matrix-valued covariance functions for stability.
• * Utilizing product kernels for accurate implicit methods and sum kernels for fast explicit methods.

Main Results:

• * Demonstrated comparable performance to spectral bases and neural networks for symplectic flow maps.
• * Showcased applicability to Poincaré maps and accurate representation of chaotic diffusion.
• * Achieved substantial performance gains in learning Hamiltonian functions from time-series data.

Conclusions:

• * The GP-based approach provides stable and accurate emulators for dynamical systems.
• * This method is effective for applications in accelerators, plasma physics, and learning system Hamiltonians.
• * Offers a powerful data-driven alternative for simulating complex physical systems.