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Learning of Hamiltonian Dynamics with Reproducing Kernel Hilbert Spaces

Torbjørn Smith, Olav Egeland

TL;DR

The simulations show that the learned dynamics reflect the energy-preservation of the Hamiltonian dynamics, and that the restriction to symplectic and odd dynamics gives improved accuracy over a large domain of the phase space.

Abstract

This paper presents a method for learning Hamiltonian dynamics from a limited set of data points. The Hamiltonian vector field is found by regularized optimization over a reproducing kernel Hilbert space of vector fields that are inherently Hamiltonian, and where the vector field is required to be odd or even. This is done with a symplectic kernel, and it is shown how this symplectic kernel can be modified to be odd or even. The performance of the method is validated in simulations for two Hamiltonian systems. The simulations show that the learned dynamics reflect the energy-preservation of the Hamiltonian dynamics, and that the restriction to symplectic and odd dynamics gives improved accuracy over a large domain of the phase space.

Learning of Hamiltonian Dynamics with Reproducing Kernel Hilbert Spaces

TL;DR

The simulations show that the learned dynamics reflect the energy-preservation of the Hamiltonian dynamics, and that the restriction to symplectic and odd dynamics gives improved accuracy over a large domain of the phase space.

Abstract

This paper presents a method for learning Hamiltonian dynamics from a limited set of data points. The Hamiltonian vector field is found by regularized optimization over a reproducing kernel Hilbert space of vector fields that are inherently Hamiltonian, and where the vector field is required to be odd or even. This is done with a symplectic kernel, and it is shown how this symplectic kernel can be modified to be odd or even. The performance of the method is validated in simulations for two Hamiltonian systems. The simulations show that the learned dynamics reflect the energy-preservation of the Hamiltonian dynamics, and that the restriction to symplectic and odd dynamics gives improved accuracy over a large domain of the phase space.
Paper Structure (20 sections, 62 equations, 4 figures, 2 tables)

This paper contains 20 sections, 62 equations, 4 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM FORMULATION
  3. PRELIMINARIES
  4. Reproducing kernel Hilbert space
  5. Learning dynamical systems with RKHS
  6. Hamiltonian dynamics
  7. Hamiltonian dynamics in the phase space
  8. Symplectic property of Hamiltonian dynamics
  9. METHOD
  10. Gaussian kernel
  11. Odd kernel
  12. Curl-free kernel
  13. Symplectic kernel
  14. Odd symplectic kernel
  15. EXPERIMENTS
  16. ...and 5 more sections

Figures (4)

  • Figure 1: Stream and trajectory plots for the harmonic oscillator and extracted data set, and the resulting learned models using the separable Gaussian kernel and the odd symplectic kernel.
  • Figure 2: Stream and trajectory plots for the simple pendulum and extracted data set, and the resulting learned models using the separable Gaussian kernel and the odd symplectic kernel.
  • Figure 3: Comparison of the two learned models against the harmonic oscillator system, using the test trajectory.
  • Figure 4: Comparison of the two learned models against the simple pendulum system, using the test trajectory.