# Hamiltonian neural networks for solving equations of motion

arXiv: https://arxiv.org/abs/2001.11107 · HTML: https://mlsanity.com/paper/2001.11107/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2020.

## Built on by (most cited first, 30 of 33)

- [One-Shot Transfer Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2110.11286/) (2021)
- [Transfer Learning with Physics-Informed Neural Networks for Efficient Simulation of Branched Flows](https://mlsanity.com/paper/2211.00214/) (2022)
- [Thermodynamics of learning physical phenomena](https://mlsanity.com/paper/2207.12749/) (2022)
- [Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftings](https://mlsanity.com/paper/2308.01084/) (2023)
- [Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space](https://mlsanity.com/paper/2501.12116/) (2025)
- [Physics-informed neural network solves minimal surfaces in curved spacetime](https://mlsanity.com/paper/2509.10866/) (2025)
- [A comparison of Single- and Double-generator formalisms for Thermodynamics-Informed Neural Networks](https://mlsanity.com/paper/2404.01060/) (2024)
- [Symplectic Neural Flows for Modeling and Discovery](https://mlsanity.com/paper/2412.16787/) (2024)
- [Semi-Explicit Neural DAEs: Learning Long-Horizon Dynamical Systems with Algebraic Constraints](https://mlsanity.com/paper/2505.20515/) (2025)
- [RcTorch: a PyTorch Reservoir Computing Package with Automated Hyper-Parameter Optimization](https://mlsanity.com/paper/2207.05870/) (2022)
- [Data-Driven Discovery of Conservation Laws from Trajectories via Neural Deflation](https://mlsanity.com/paper/2410.05445/) (2024)
- [Bayesian Inference with Latent Hamiltonian Neural Networks](https://mlsanity.com/paper/2208.06120/) (2022)
- [First principles physics-informed neural network for quantum wavefunctions and eigenvalue surfaces](https://mlsanity.com/paper/2211.04607/) (2022)
- [Lagrangian neural networks for nonholonomic mechanics](https://mlsanity.com/paper/2411.00110/) (2024)
- [Discovering Phase Space Structure in Learned Hamiltonian Systems](https://mlsanity.com/paper/2509.23003/) (2025)
- [Enhancing Neural Network Differential Equation Solvers](https://mlsanity.com/paper/2301.13146/) (2022)
- [Physics-Informed Neural Networks for Discovering Localised Eigenstates in Disordered Media](https://mlsanity.com/paper/2305.06802/) (2023)
- [Iterative Magnitude Pruning as a Renormalisation Group: A Study in The Context of The Lottery Ticket Hypothesis](https://mlsanity.com/paper/2308.03128/) (2023)
- [Separable Hamiltonian Neural Networks](https://mlsanity.com/paper/2309.01069/) (2023)
- [Hamiltonian Neural Networks approach to fuzzball geodesics](https://mlsanity.com/paper/2502.20881/) (2025)
- [Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structures](https://mlsanity.com/paper/2508.01854/) (2025)
- [Multi-resolution Physics-Aware Recurrent Convolutional Neural Network for Complex Flows](https://mlsanity.com/paper/2512.06031/) (2025)
- [Symmetry-Constrained Language-Guided Program Synthesis for Discovering Governing Equations from Noisy and Partial Observations](https://mlsanity.com/paper/2603.06869/) (2026)
- [Physics-Informed Machine Learning of Dynamical Systems for Efficient Bayesian Inference](https://mlsanity.com/paper/2209.09349/) (2022)
- [Optimization-Informed Neural Networks](https://mlsanity.com/paper/2210.02113/) (2022)
- [Transferability of Winning Lottery Tickets in Neural Network Differential Equation Solvers](https://mlsanity.com/paper/2306.09863/) (2023)
- [Equivariant graph convolutional neural networks for the representation of homogenized anisotropic microstructural mechanical response](https://mlsanity.com/paper/2404.17584/) (2024)
- [GradINN: Gradient Informed Neural Network](https://mlsanity.com/paper/2409.01914/) (2024)
- [UP-dROM : Uncertainty-Aware and Parametrised dynamic Reduced-Order Model, application to unsteady flows](https://mlsanity.com/paper/2503.23236/) (2025)
- [Hamiltonian Neural PDE Solvers through Functional Approximation](https://mlsanity.com/paper/2505.13275/) (2025)

## Abstract (arXiv)

There has been a wave of interest in applying machine learning to study dynamical systems. We present a Hamiltonian neural network that solves the differential equations that govern dynamical systems. This is an equation-driven machine learning method where the optimization process of the network depends solely on the predicted functions without using any ground truth data. The model learns solutions that satisfy, up to an arbitrarily small error, Hamilton's equations and, therefore, conserve the Hamiltonian invariants. The choice of an appropriate activation function drastically improves the predictability of the network. Moreover, an error analysis is derived and states that the numerical errors depend on the overall network performance. The Hamiltonian network is then employed to solve the equations for the nonlinear oscillator and the chaotic Henon-Heiles dynamical system. In both systems, a symplectic Euler integrator requires two orders more evaluation points than the Hamiltonian network in order to achieve the same order of the numerical error in the predicted phase space trajectories.