# Hamiltonian Graph Networks with ODE Integrators

arXiv: https://arxiv.org/abs/1909.12790 · HTML: https://mlsanity.com/paper/1909.12790/

## Key facts

- Cited by 108 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Sep 2019.

## Built on by (most cited first, 30 of 108)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
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- [A Survey on Graph Neural Networks for Time Series: Forecasting, Classification, Imputation, and Anomaly Detection](https://mlsanity.com/paper/2307.03759/) (2023)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [gradSim: Differentiable simulation for system identification and visuomotor control](https://mlsanity.com/paper/2104.02646/) (2021)
- [EqMotion: Equivariant Multi-agent Motion Prediction with Invariant Interaction Reasoning](https://mlsanity.com/paper/2303.10876/) (2023)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Equivariant Graph Mechanics Networks with Constraints](https://mlsanity.com/paper/2203.06442/) (2022)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [Geometrically Equivariant Graph Neural Networks: A Survey](https://mlsanity.com/paper/2202.07230/) (2022)
- [Machine learning structure preserving brackets for forecasting irreversible processes](https://mlsanity.com/paper/2106.12619/) (2021)
- [One-Shot Transfer Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2110.11286/) (2021)
- [Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separately](https://mlsanity.com/paper/2201.10085/) (2022)
- [MALI: A memory efficient and reverse accurate integrator for Neural ODEs](https://mlsanity.com/paper/2102.04668/) (2021)
- [Simulating Continuum Mechanics with Multi-Scale Graph Neural Networks](https://mlsanity.com/paper/2106.04900/) (2021)
- [Learning Physical Dynamics with Subequivariant Graph Neural Networks](https://mlsanity.com/paper/2210.06876/) (2022)
- [Equivariant Graph Neural Operator for Modeling 3D Dynamics](https://mlsanity.com/paper/2401.11037/) (2024)
- [Thermodynamics-informed graph neural networks](https://mlsanity.com/paper/2203.01874/) (2022)
- [Constructing Neural Network-Based Models for Simulating Dynamical Systems](https://mlsanity.com/paper/2111.01495/) (2021)
- [ACMP: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networks](https://mlsanity.com/paper/2206.05437/) (2022)
- [Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics](https://mlsanity.com/paper/2210.06036/) (2022)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [Constraint-based graph network simulator](https://mlsanity.com/paper/2112.09161/) (2021)
- [SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biases](https://mlsanity.com/paper/2308.13212/) (2023)
- [Which priors matter? Benchmarking models for learning latent dynamics](https://mlsanity.com/paper/2111.05458/) (2021)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)

## Abstract (arXiv)

We introduce an approach for imposing physically informed inductive biases in learned simulation models. We combine graph networks with a differentiable ordinary differential equation integrator as a mechanism for predicting future states, and a Hamiltonian as an internal representation. We find that our approach outperforms baselines without these biases in terms of predictive accuracy, energy accuracy, and zero-shot generalization to time-step sizes and integrator orders not experienced during training. This advances the state-of-the-art of learned simulation, and in principle is applicable beyond physical domains.