# Data-efficient Kernel Methods for Learning Hamiltonian Systems

arXiv: https://arxiv.org/abs/2509.17154 · HTML: https://mlsanity.com/paper/2509.17154/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 71 references that are in the map.
- First posted on arXiv 21 Sep 2025.

## Builds on (most cited first, 27 of 27)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Computational Graph Completion](https://mlsanity.com/paper/2110.10323/) (2021)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [Gaussian processes meet NeuralODEs: A Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy data](https://mlsanity.com/paper/2103.03385/) (2021)
- [Learning reversible symplectic dynamics](https://mlsanity.com/paper/2204.12323/) (2022)
- [Structure-preserving Gaussian Process Dynamics](https://mlsanity.com/paper/2102.01606/) (2021)
- [A Structure-Preserving Kernel Method for Learning Hamiltonian Systems](https://mlsanity.com/paper/2403.10070/) (2024)
- [Gaussian Process Latent Force Models for Learning and Stochastic Control of Physical Systems](https://mlsanity.com/paper/1709.05409/) (2017)
- [Recognition Models to Learn Dynamics from Partial Observations with Neural ODEs](https://mlsanity.com/paper/2205.12550/) (2022)
- [Symbolic Recovery of Differential Equations: The Identifiability Problem](https://mlsanity.com/paper/2210.08342/) (2022)
- [A Kernel Approach for PDE Discovery and Operator Learning](https://mlsanity.com/paper/2210.08140/) (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)
- [Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approach](https://mlsanity.com/paper/2408.06465/) (2024)
- [Structure-Preserving Learning Using Gaussian Processes and Variational Integrators](https://mlsanity.com/paper/2112.05451/) (2021)
- [Comment on "Machine learning conservation laws from differential equations"](https://mlsanity.com/paper/2404.02896/) (2024)

## Built on by (most cited first, 2 of 2)

- [Physics-Informed Neural Networks for Joint Source and Parameter Estimation in Advection-Diffusion Equations](https://mlsanity.com/paper/2512.07755/) (2025)
- [Adaptive Kernel Selection for Kernelized Diffusion Maps](https://mlsanity.com/paper/2604.18402/) (2026)

## Abstract (arXiv)

Hamiltonian dynamics describe a wide range of physical systems. As such, data-driven simulations of Hamiltonian systems are important for many scientific and engineering problems. In this work, we propose kernel-based methods for identifying and forecasting Hamiltonian systems directly from trajectory data. We present two approaches: a 2-step method that reconstructs trajectories before learning the Hamiltonian, and a 1-step method that jointly infers both. Across several benchmark systems, including mass-spring dynamics, a nonlinear pendulum, and the Henon-Heiles system, we demonstrate that our framework achieves accurate, data-efficient predictions and outperforms 2-step kernel-based baselines, particularly in scarce-data regimes, while preserving the Hamiltonian structure. Moreover, we prove a priori error estimates, ensuring reliability of the learned models. We also provide a more general, problem-agnostic numerical framework that goes beyond Hamiltonian systems and can be used for data-driven learning of arbitrary dynamical systems.