# Symplectic Gaussian Process Regression of Hamiltonian Flow Maps

arXiv: https://arxiv.org/abs/2009.05569 · HTML: https://mlsanity.com/paper/2009.05569/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Sep 2020.

## Built on by (most cited first, 17 of 17)

- [Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Prior](https://mlsanity.com/paper/2305.09017/) (2023)
- [Structure-preserving Gaussian Process Dynamics](https://mlsanity.com/paper/2102.01606/) (2021)
- [Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systems](https://mlsanity.com/paper/2211.16006/) (2022)
- [Approximation of nearly-periodic symplectic maps via structure-preserving neural networks](https://mlsanity.com/paper/2210.05087/) (2022)
- [Hamiltonian Neural Networks with Automatic Symmetry Detection](https://mlsanity.com/paper/2301.07928/) (2023)
- [Learning Energy Conserving Dynamics Efficiently with Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2303.01925/) (2023)
- [SyMetric: Measuring the Quality of Learnt Hamiltonian Dynamics Inferred from Vision](https://mlsanity.com/paper/2111.05986/) (2021)
- [Physically Consistent Learning of Conservative Lagrangian Systems with Gaussian Processes](https://mlsanity.com/paper/2206.12272/) (2022)
- [Exact Inference for Continuous-Time Gaussian Process Dynamics](https://mlsanity.com/paper/2309.02351/) (2023)
- [Learning Hamiltonian Dynamics with Reproducing Kernel Hilbert Spaces and Random Features](https://mlsanity.com/paper/2404.07703/) (2024)
- [Data-driven identification of port-Hamiltonian DAE systems by Gaussian processes](https://mlsanity.com/paper/2406.18726/) (2024)
- [Learning the action for long-time-step simulations of molecular dynamics](https://mlsanity.com/paper/2508.01068/) (2025)
- [Learning Generalized Hamiltonian Dynamics with Stability from Noisy Trajectory Data](https://mlsanity.com/paper/2509.07280/) (2025)
- [Learning Dynamics from Input-Output Data with Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2511.05330/) (2025)
- [Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs](https://mlsanity.com/paper/2402.09018/) (2024)
- [Structure-Preserving Gaussian Processes Via Discrete Euler-Lagrange Equations](https://mlsanity.com/paper/2605.06246/) (2026)
- [Gaussian process learning with flow map refinement for parameter estimation in dynamical systems](https://mlsanity.com/paper/2608.22324/) (2026)

## Abstract (arXiv)

We present an approach to construct appropriate and efficient emulators for Hamiltonian flow maps. Intended future applications are long-term tracing of fast charged particles in accelerators and magnetic plasma confinement configurations. The method is based on multi-output Gaussian process regression on scattered training data. To obtain long-term stability the symplectic property is enforced via the choice of the matrix-valued covariance function. Based on earlier work on spline interpolation we observe derivatives of the generating function of a canonical transformation. A product kernel produces an accurate implicit method, whereas a sum kernel results in a fast explicit method from this approach. Both correspond to a symplectic Euler method in terms of numerical integration. These methods are applied to the pendulum and the H'enon-Heiles system and results compared to an symmetric regression with orthogonal polynomials. In the limit of small mapping times, the Hamiltonian function can be identified with a part of the generating function and thereby learned from observed time-series data of the system's evolution. Besides comparable performance of implicit kernel and spectral regression for symplectic maps, we demonstrate a substantial increase in performance for learning the Hamiltonian function compared to existing approaches.