# Physics-Informed Learning Using Hamiltonian Neural Networks with Output Error Noise Models

arXiv: https://arxiv.org/abs/2305.01338 · HTML: https://mlsanity.com/paper/2305.01338/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 17 references that are in the map.
- First posted on arXiv 2 May 2023.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [Nonlinear System Identification: A User-Oriented Roadmap](https://mlsanity.com/paper/1902.00683/) (2019)
- [Machine learning structure preserving brackets for forecasting irreversible processes](https://mlsanity.com/paper/2106.12619/) (2021)
- [Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separately](https://mlsanity.com/paper/2201.10085/) (2022)
- [Nonseparable Symplectic Neural Networks](https://mlsanity.com/paper/2010.12636/) (2020)
- [Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm](https://mlsanity.com/paper/1910.02071/) (2019)
- [Learning Neural Hamiltonian Dynamics: A Methodological Overview](https://mlsanity.com/paper/2203.00128/) (2022)
- [Bayesian Identification of Nonseparable Hamiltonian Systems Using Stochastic Dynamic Models](https://mlsanity.com/paper/2209.07646/) (2022)

## Built on by (most cited first, 5 of 5)

- [Meta-Learning for Physically-Constrained Neural System Identification](https://mlsanity.com/paper/2501.06167/) (2025)
- [State-Space Kolmogorov Arnold Networks for Interpretable Nonlinear System Identification](https://mlsanity.com/paper/2506.16392/) (2025)
- [Common pitfalls to avoid while using multiobjective optimization in machine learning](https://mlsanity.com/paper/2405.01480/) (2024)
- [Manifold meta-learning for reduced-complexity neural system identification](https://mlsanity.com/paper/2504.11811/) (2025)
- [Learning Physically Consistent Lagrangian Control Models Without Acceleration Measurements](https://mlsanity.com/paper/2512.03035/) (2025)

## Abstract (arXiv)

In order to make data-driven models of physical systems interpretable and reliable, it is essential to include prior physical knowledge in the modeling framework. Hamiltonian Neural Networks (HNNs) implement Hamiltonian theory in deep learning and form a comprehensive framework for modeling autonomous energy-conservative systems. Despite being suitable to estimate a wide range of physical system behavior from data, classical HNNs are restricted to systems without inputs and require noiseless state measurements and information on the derivative of the state to be available. To address these challenges, this paper introduces an Output Error Hamiltonian Neural Network (OE-HNN) modeling approach to address the modeling of physical systems with inputs and noisy state measurements. Furthermore, it does not require the state derivatives to be known. Instead, the OE-HNN utilizes an ODE-solver embedded in the training process, which enables the OE-HNN to learn the dynamics from noisy state measurements. In addition, extending HNNs based on the generalized Hamiltonian theory enables to include external inputs into the framework which are important for engineering applications. We demonstrate via simulation examples that the proposed OE-HNNs results in superior modeling performance compared to classical HNNs.