# Lagrangian neural ODEs: Measuring the existence of a Lagrangian with Helmholtz metrics

arXiv: https://arxiv.org/abs/2510.06367 · HTML: https://mlsanity.com/paper/2510.06367/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 10 references that are in the map.
- First posted on arXiv 7 Oct 2025.

## Builds on (most cited first, 7 of 7)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Pareto Multi-Task Learning](https://mlsanity.com/paper/1912.12854/) (2019)
- [On Second Order Behaviour in Augmented Neural ODEs](https://mlsanity.com/paper/2006.07220/) (2020)

## Abstract (arXiv)

Neural ODEs are a widely used, powerful machine learning technique in particular for physics. However, not every solution is physical in that it is an Euler-Lagrange equation. We present Helmholtz metrics to quantify this resemblance for a given ODE and demonstrate their capabilities on several fundamental systems with noise. We combine them with a second order neural ODE to form a Lagrangian neural ODE, which allows to learn Euler-Lagrange equations in a direct fashion and with zero additional inference cost. We demonstrate that, using only positional data, they can distinguish Lagrangian and non-Lagrangian systems and improve the neural ODE solutions.