Symplectic integration of learned Hamiltonian systems

Area of Science:

• Computational Physics
• Applied Mathematics
• Machine Learning

Background:

• Hamiltonian systems are crucial in classical mechanics, plasma physics, and sampling.
• Predicting Hamiltonian dynamics requires incorporating prior knowledge of system structure.
• Current methods involve learning the Hamiltonian and using symplectic integrators, which introduce approximation and discretization errors.

Purpose of the Study:

• To develop a method for learning Hamiltonian structures directly from trajectory observations.
• To eliminate the need for separate Hamiltonian data approximation steps.
• To compensate for and eliminate discretization errors in predicting Hamiltonian dynamics.

Main Methods:

• Learning an inverse modified Hamiltonian structure directly from observed data.
• Adapting the learned structure to a geometric integrator.
• Utilizing Gaussian processes for the learning technique.

Main Results:

• Successfully learned an inverse modified Hamiltonian structure directly from observations.
• Avoided the separate approximation step for Hamiltonian data.
• Demonstrated the elimination of discretization error by compensating for it.

Conclusions:

• The proposed method accurately predicts Hamiltonian dynamics by directly learning the system's structure.
• This approach enhances prediction accuracy by removing approximation and discretization errors.
• The technique offers a more efficient and accurate way to analyze Hamiltonian systems using observational data.