# Learning Passive Continuous-Time Dynamics with Multistep Port-Hamiltonian Gaussian Processes

arXiv: https://arxiv.org/abs/2510.00384 · HTML: https://mlsanity.com/paper/2510.00384/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 34 references that are in the map.
- First posted on arXiv 1 Oct 2025.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Control Barrier Functions for Unknown Nonlinear Systems using Gaussian Processes](https://mlsanity.com/paper/2010.05818/) (2020)
- [Compositional Learning of Dynamical System Models Using Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2212.00893/) (2022)
- [Gaussian Process Port-Hamiltonian Systems: Bayesian Learning with Physics Prior](https://mlsanity.com/paper/2305.09017/) (2023)
- [Exact Inference for Continuous-Time Gaussian Process Dynamics](https://mlsanity.com/paper/2309.02351/) (2023)
- [Data-driven Bayesian Control of Port-Hamiltonian Systems](https://mlsanity.com/paper/2309.04678/) (2023)
- [Data-driven identification of port-Hamiltonian DAE systems by Gaussian processes](https://mlsanity.com/paper/2406.18726/) (2024)
- [Spectral Flow Learning Theory: Finite-Sample Guarantees for Vector-Field Identification](https://mlsanity.com/paper/2509.25000/) (2025)

## Abstract (arXiv)

We propose the multistep port-Hamiltonian Gaussian process (MS-PHS GP) to learn physically consistent continuous-time dynamics and a posterior over the Hamiltonian from noisy, irregularly-sampled trajectories. By placing a GP prior on the Hamiltonian surface H and encoding variable-step multistep integrator constraints as finite linear functionals, MS-PHS GP enables closed-form conditioning of both the vector field and the Hamiltonian surface without latent states, while enforcing energy balance and passivity by design. We state a finite-sample vector-field bound that separates the estimation and variable-step discretization terms. Lastly, we demonstrate improved vector-field recovery and well-calibrated Hamiltonian uncertainty on mass-spring, Van der Pol, and Duffing benchmarks.