What is it about?

Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system’s Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data needs to be approximated based on the trajectory observations. Moreover, the numerical integrator introduces an additional discretisation error. In this paper, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data is avoided. The inverse modified data compensates for the discretisation error such that the discretisation error is eliminated. The technique is developed for Gaussian Processes.

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Why is it important?

Combining trajectory data with prior knowledge about structural properties of a dynamical system is known to greatly improve predictions of the system’s motions. The article introduces the new technique Symplectic Shadow Integration, which incorporates modified structures into learned models. The modified structures compensate discretisation errors which limit the accuracy of existing approaches. Symplectic Shadow Integrators achieve extremely high energy accuracy (down to random walk) because they compensate for discretisation errors.


I have worked a lot with backward error analysis to analyse discretisation errors of numerical methods. I was very pleased to develop a framework which puts this established technique into new use in the context of machine learning.

Christian Offen
Universitat Paderborn

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This page is a summary of: Symplectic integration of learned Hamiltonian systems, Chaos An Interdisciplinary Journal of Nonlinear Science, January 2022, American Institute of Physics, DOI: 10.1063/5.0065913.
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