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Learning dissipative Hamiltonian dynamics with reproducing kernel Hilbert spaces and random Fourier features

Torbjørn Smith, Olav Egeland

TL;DR

This work tackles learning dissipative Hamiltonian dynamics from limited, noisy data while enforcing physical structure. It introduces a Helmholtz-decomposition-based RKHS framework that independently learns a symplectic part via a curl-free kernel and a dissipative part via a curl-free kernel, with both components endowed with odd symmetry and approximated by random Fourier features for efficiency, yielding f(x) = f_s(x) + f_d(x) and f_s(x) = J ∇H(x), f_d(x) = −R ∇H(x). Key contributions include the construction of odd curl-free and odd symplectic kernels, the RFF-based Helmholtz learning algorithm, and empirical validation showing improved predictive accuracy over a Gaussian-separable kernel baseline on two mechanical systems. The results demonstrate data-efficient, physically-consistent modeling of dissipative Hamiltonian systems with potential impact on control and robotics applications where energy dissipation cannot be neglected.

Abstract

This paper presents a new method for learning dissipative Hamiltonian dynamics from a limited and noisy dataset. The method uses the Helmholtz decomposition to learn a vector field as the sum of a symplectic and a dissipative vector field. The two vector fields are learned using two reproducing kernel Hilbert spaces, defined by a symplectic and a curl-free kernel, where the kernels are specialized to enforce odd symmetry. Random Fourier features are used to approximate the kernels to reduce the dimension of the optimization problem. The performance of the method is validated in simulations for two dissipative Hamiltonian systems, and it is shown that the method improves predictive accuracy significantly compared to a method where a Gaussian separable kernel is used.

Learning dissipative Hamiltonian dynamics with reproducing kernel Hilbert spaces and random Fourier features

TL;DR

This work tackles learning dissipative Hamiltonian dynamics from limited, noisy data while enforcing physical structure. It introduces a Helmholtz-decomposition-based RKHS framework that independently learns a symplectic part via a curl-free kernel and a dissipative part via a curl-free kernel, with both components endowed with odd symmetry and approximated by random Fourier features for efficiency, yielding f(x) = f_s(x) + f_d(x) and f_s(x) = J ∇H(x), f_d(x) = −R ∇H(x). Key contributions include the construction of odd curl-free and odd symplectic kernels, the RFF-based Helmholtz learning algorithm, and empirical validation showing improved predictive accuracy over a Gaussian-separable kernel baseline on two mechanical systems. The results demonstrate data-efficient, physically-consistent modeling of dissipative Hamiltonian systems with potential impact on control and robotics applications where energy dissipation cannot be neglected.

Abstract

This paper presents a new method for learning dissipative Hamiltonian dynamics from a limited and noisy dataset. The method uses the Helmholtz decomposition to learn a vector field as the sum of a symplectic and a dissipative vector field. The two vector fields are learned using two reproducing kernel Hilbert spaces, defined by a symplectic and a curl-free kernel, where the kernels are specialized to enforce odd symmetry. Random Fourier features are used to approximate the kernels to reduce the dimension of the optimization problem. The performance of the method is validated in simulations for two dissipative Hamiltonian systems, and it is shown that the method improves predictive accuracy significantly compared to a method where a Gaussian separable kernel is used.

Paper Structure

This paper contains 19 sections, 42 equations, 2 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Reproducing kernel Hilbert space
  4. Learning dynamical systems with RKHS
  5. Random Fourier features
  6. Hamiltonian dynamics
  7. Dissipative Hamiltonian dynamics
  8. METHOD
  9. Odd kernel
  10. Curl-free kernels and divergence-free kernels
  11. Symplectic kernel as divergence-free kernel
  12. Odd curl-free and odd symplectic kernel
  13. RFF approximation of the odd curl-free and odd symplectic kernels
  14. Solution of the Helmholtz decomposition
  15. EXPERIMENTS
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Stream and trajectory plots for the mass-spring-damper and extracted data set, and the resulting learned models using the Gaussian separable kernel and the Helmholtz model.
  • Figure 2: Stream and trajectory plots for the damped pendulum and extracted data set, and the resulting learned models using the Gaussian separable kernel and the Helmholtz model.