# Lagrangian Neural Network with Differentiable Symmetries and Relational Inductive Bias

arXiv: https://arxiv.org/abs/2110.03266 · HTML: https://mlsanity.com/paper/2110.03266/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 15 references that are in the map.
- First posted on arXiv 7 Oct 2021.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [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)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data](https://mlsanity.com/paper/2012.02334/) (2020)
- [Unsupervised Learning of Lagrangian Dynamics from Images for Prediction and Control](https://mlsanity.com/paper/2007.01926/) (2020)
- [Modeling System Dynamics with Physics-Informed Neural Networks Based on Lagrangian Mechanics](https://mlsanity.com/paper/2005.14617/) (2020)

## Built on by (most cited first, 4 of 4)

- [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)
- [A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic](https://mlsanity.com/paper/2204.06362/) (2022)
- [Recovering Irreducible Three-Body Interactions with Lagrangian Neural Networks](https://mlsanity.com/paper/2610.13292/) (2026)

## Abstract (arXiv)

Realistic models of physical world rely on differentiable symmetries that, in turn, correspond to conservation laws. Recent works on Lagrangian and Hamiltonian neural networks show that the underlying symmetries of a system can be easily learned by a neural network when provided with an appropriate inductive bias. However, these models still suffer from issues such as inability to generalize to arbitrary system sizes, poor interpretability, and most importantly, inability to learn translational and rotational symmetries, which lead to the conservation laws of linear and angular momentum, respectively. Here, we present a momentum conserving Lagrangian neural network (MCLNN) that learns the Lagrangian of a system, while also preserving the translational and rotational symmetries. We test our approach on linear and non-linear spring systems, and a gravitational system, demonstrating the energy and momentum conservation. We also show that the model developed can generalize to systems of any arbitrary size. Finally, we discuss the interpretability of the MCLNN, which directly provides physical insights into the interactions of multi-particle systems.