# Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints

arXiv: https://arxiv.org/abs/2010.13581 · HTML: https://mlsanity.com/paper/2010.13581/

## Key facts

- Cited by 105 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Oct 2020.

## Built on by (most cited first, 30 of 105)

- [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)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modeling](https://mlsanity.com/paper/2109.06407/) (2021)
- [Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Control](https://mlsanity.com/paper/2106.12782/) (2021)
- [Extending Lagrangian and Hamiltonian Neural Networks with Differentiable Contact Models](https://mlsanity.com/paper/2102.06794/) (2021)
- [Constructing Neural Network-Based Models for Simulating Dynamical Systems](https://mlsanity.com/paper/2111.01495/) (2021)
- [Deconstructing the Inductive Biases of Hamiltonian Neural Networks](https://mlsanity.com/paper/2202.04836/) (2022)
- [User-defined Event Sampling and Uncertainty Quantification in Diffusion Models for Physical Dynamical Systems](https://mlsanity.com/paper/2306.07526/) (2023)
- [Self-Supervised Learning with Lie Symmetries for Partial Differential Equations](https://mlsanity.com/paper/2307.05432/) (2023)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [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)
- [Noether Networks: Meta-Learning Useful Conserved Quantities](https://mlsanity.com/paper/2112.03321/) (2021)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [Learning Articulated Rigid Body Dynamics with Lagrangian Graph Neural Network](https://mlsanity.com/paper/2209.11588/) (2022)
- [Unravelling the Performance of Physics-informed Graph Neural Networks for Dynamical Systems](https://mlsanity.com/paper/2211.05520/) (2022)
- [Combining Physics and Deep Learning to learn Continuous-Time Dynamics Models](https://mlsanity.com/paper/2110.01894/) (2021)
- [Climate Modeling with Neural Diffusion Equations](https://mlsanity.com/paper/2111.06011/) (2021)
- [Enhancing the Inductive Biases of Graph Neural ODE for Modeling Dynamical Systems](https://mlsanity.com/paper/2209.10740/) (2022)
- [Pseudo-Hamiltonian Neural Networks with State-Dependent External Forces](https://mlsanity.com/paper/2206.02660/) (2022)
- [Compositional Learning of Dynamical System Models Using Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2212.00893/) (2022)
- [Physics-Informed Model-Based Reinforcement Learning](https://mlsanity.com/paper/2212.02179/) (2022)
- [Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraints](https://mlsanity.com/paper/2306.09739/) (2023)
- [Learning Hamiltonians of constrained mechanical systems](https://mlsanity.com/paper/2201.13254/) (2022)
- [Homotopy-based training of NeuralODEs for accurate dynamics discovery](https://mlsanity.com/paper/2210.01407/) (2022)
- [Forecasting the outcome of spintronic experiments with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2108.02318/) (2021)
- [Applying Machine Learning to Study Fluid Mechanics](https://mlsanity.com/paper/2110.02083/) (2021)
- [Taylor-Lagrange Neural Ordinary Differential Equations: Toward Fast Training and Evaluation of Neural ODEs](https://mlsanity.com/paper/2201.05715/) (2022)
- [Learning the Dynamics of Particle-based Systems with Lagrangian Graph Neural Networks](https://mlsanity.com/paper/2209.01476/) (2022)

## Abstract (arXiv)

Reasoning about the physical world requires models that are endowed with the right inductive biases to learn the underlying dynamics. Recent works improve generalization for predicting trajectories by learning the Hamiltonian or Lagrangian of a system rather than the differential equations directly. While these methods encode the constraints of the systems using generalized coordinates, we show that embedding the system into Cartesian coordinates and enforcing the constraints explicitly with Lagrange multipliers dramatically simplifies the learning problem. We introduce a series of challenging chaotic and extended-body systems, including systems with N-pendulums, spring coupling, magnetic fields, rigid rotors, and gyroscopes, to push the limits of current approaches. Our experiments show that Cartesian coordinates with explicit constraints lead to a 100x improvement in accuracy and data efficiency.