# Learning Neural Hamiltonian Dynamics: A Methodological Overview

arXiv: https://arxiv.org/abs/2203.00128 · HTML: https://mlsanity.com/paper/2203.00128/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 56 references that are in the map.
- First posted on arXiv 28 Feb 2022.

## Builds on (most cited first, 29 of 29)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (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)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [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)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Differentiable Game Mechanics](https://mlsanity.com/paper/1905.04926/) (2019)
- [Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps](https://mlsanity.com/paper/2103.05632/) (2021)
- [Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data](https://mlsanity.com/paper/2012.02334/) (2020)
- [Variational Integrator Networks for Physically Structured Embeddings](https://mlsanity.com/paper/1910.09349/) (2019)
- [Sparse Symplectically Integrated Neural Networks](https://mlsanity.com/paper/2006.12972/) (2020)
- [Learning Hamiltonians of constrained mechanical systems](https://mlsanity.com/paper/2201.13254/) (2022)
- [HamNet: Conformation-Guided Molecular Representation with Hamiltonian Neural Networks](https://mlsanity.com/paper/2105.03688/) (2021)
- [Symplectic Adjoint Method for Exact Gradient of Neural ODE with Minimal Memory](https://mlsanity.com/paper/2102.09750/) (2021)
- [Identifying Physical Law of Hamiltonian Systems via Meta-Learning](https://mlsanity.com/paper/2102.11544/) (2021)
- [Weak Form Generalized Hamiltonian Learning](https://mlsanity.com/paper/2104.05096/) (2021)
- [Active Learning of Quantum System Hamiltonians yields Query Advantage](https://mlsanity.com/paper/2112.14553/) (2021)
- [Neural Symplectic Integrator with Hamiltonian Inductive Bias for the Gravitational N-body Problem](https://mlsanity.com/paper/2111.15631/) (2021)

## Built on by (most cited first, 8 of 8)

- [Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraints](https://mlsanity.com/paper/2306.09739/) (2023)
- [Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systems](https://mlsanity.com/paper/2211.01873/) (2022)
- [Thermodynamics of learning physical phenomena](https://mlsanity.com/paper/2207.12749/) (2022)
- [Physics-Informed Learning Using Hamiltonian Neural Networks with Output Error Noise Models](https://mlsanity.com/paper/2305.01338/) (2023)
- [SlotPi: Physics-informed Object-centric Reasoning Models](https://mlsanity.com/paper/2506.10778/) (2025)
- [Separable Hamiltonian Neural Networks](https://mlsanity.com/paper/2309.01069/) (2023)
- [Phase autoencoder for limit-cycle oscillators](https://mlsanity.com/paper/2403.06992/) (2024)
- [Neural Hamilton: Can A.I. Understand Hamiltonian Mechanics?](https://mlsanity.com/paper/2410.20951/) (2024)

## Abstract (arXiv)

The past few years have witnessed an increased interest in learning Hamiltonian dynamics in deep learning frameworks. As an inductive bias based on physical laws, Hamiltonian dynamics endow neural networks with accurate long-term prediction, interpretability, and data-efficient learning. However, Hamiltonian dynamics also bring energy conservation or dissipation assumptions on the input data and additional computational overhead. In this paper, we systematically survey recently proposed Hamiltonian neural network models, with a special emphasis on methodologies. In general, we discuss the major contributions of these models, and compare them in four overlapping directions: 1) generalized Hamiltonian system; 2) symplectic integration, 3) generalized input form, and 4) extended problem settings. We also provide an outlook of the fundamental challenges and emerging opportunities in this area.