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MPI CyberMotion Simulator: Implementation of a Novel Motion Simulator to Investigate Multisensory Path Integration in Three Dimensions

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Symplectic integration of learned Hamiltonian systems.

C Offen¹, S Ober-Blöbaum¹

¹Department of Mathematics, Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany.

Chaos (Woodbury, N.Y.)

|February 2, 2022

Summary

This study introduces a new method to predict Hamiltonian dynamics from observed data. It directly learns an inverse modified Hamiltonian structure, eliminating approximation errors and improving prediction accuracy for complex systems.

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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.