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Learning Hamiltonian Dynamics with Reproducing Kernel Hilbert Spaces and Random Features

Torbjørn Smith, Olav Egeland

TL;DR

It is demonstrated that the use of an odd symplectic kernel improves prediction accuracy and data efficiency, and that the learned vector fields are Hamiltonian and exhibit the imposed odd symmetry characteristics.

Abstract

A method for learning Hamiltonian dynamics from a limited and noisy dataset is proposed. The method learns a Hamiltonian vector field on a reproducing kernel Hilbert space (RKHS) of inherently Hamiltonian vector fields, and in particular, odd Hamiltonian vector fields. This is done with a symplectic kernel, and it is shown how the kernel can be modified to an odd symplectic kernel to impose the odd symmetry. A random feature approximation is developed for the proposed odd kernel to reduce the problem size. The performance of the method is validated in simulations for three Hamiltonian systems. It is demonstrated that the use of an odd symplectic kernel improves prediction accuracy and data efficiency, and that the learned vector fields are Hamiltonian and exhibit the imposed odd symmetry characteristics.

Learning Hamiltonian Dynamics with Reproducing Kernel Hilbert Spaces and Random Features

TL;DR

It is demonstrated that the use of an odd symplectic kernel improves prediction accuracy and data efficiency, and that the learned vector fields are Hamiltonian and exhibit the imposed odd symmetry characteristics.

Abstract

A method for learning Hamiltonian dynamics from a limited and noisy dataset is proposed. The method learns a Hamiltonian vector field on a reproducing kernel Hilbert space (RKHS) of inherently Hamiltonian vector fields, and in particular, odd Hamiltonian vector fields. This is done with a symplectic kernel, and it is shown how the kernel can be modified to an odd symplectic kernel to impose the odd symmetry. A random feature approximation is developed for the proposed odd kernel to reduce the problem size. The performance of the method is validated in simulations for three Hamiltonian systems. It is demonstrated that the use of an odd symplectic kernel improves prediction accuracy and data efficiency, and that the learned vector fields are Hamiltonian and exhibit the imposed odd symmetry characteristics.
Paper Structure (27 sections, 82 equations, 5 figures, 2 tables)

This paper contains 27 sections, 82 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related work
  4. Data-driven modeling with kernels
  5. Learning Hamiltonian dynamics
  6. Paper structure
  7. Problem formulation
  8. Preliminaries
  9. Reproducing kernel Hilbert space
  10. Regularized least-squares
  11. Random Fourier features
  12. RFF for Gaussian and curl-free kernels
  13. Hamiltonian dynamics
  14. Odd reproducing kernels
  15. Odd kernel
  16. ...and 12 more sections

Figures (5)

  • Figure 1: Stream and trajectory plots for the simple pendulum and extracted data set, and the resulting learned models using the separable Gaussian kernel, symplectic kernel, SympGPR, and the odd symplectic kernel.
  • Figure 2: Comparison of the four learned models against the simple pendulum system, using the test trajectory.
  • Figure 3: Cart-pole: Mean MSE for the training set and test set over 20 different seeds for each number of initial conditions in the training set. Axes are in log-log scale
  • Figure 4: 2-link Robot: Mean MSE for the training set and test set over 20 different seeds for each number of initial conditions in the training set. Axes are in log-log scale
  • Figure 5: Trajectory MSE for the odd symplectic kernel and its RFF approximation. The error for the RFF approximation is the mean error over 50 draws of ${d = \{10,20,40,80,160,320,640,1280,2560\}}$ random features.