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

Luca Wolf, Tobias Buck, Bjoern Malte Schaefer

TL;DR

The paper addresses the question of which ODEs arise from a Lagrangian by introducing Helmholtz metrics that quantify proximity to Euler–Lagrange equations. It then couples these metrics with a second-order neural ODE to form a Lagrangian neural ODE, using a differentiable Hessian $g$ learned by a NN and a total loss $\mathcal{L}_{tot}=\mathcal{L}_R+\mathcal{L}_H$ to regularize learning. Empirical results on analytical systems (e.g., Kepler, damped oscillator) show that the Helmholtz metrics recover a Hessian close to the true Lagrangian and distinguish non-Lagrangian dynamics, while data-driven experiments demonstrate that Lagrangian-regularized models can match regression performance and improve alignment with EL structure, suggesting practical benefits for physics-informed learning and extrapolation.

Abstract

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.

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

TL;DR

The paper addresses the question of which ODEs arise from a Lagrangian by introducing Helmholtz metrics that quantify proximity to Euler–Lagrange equations. It then couples these metrics with a second-order neural ODE to form a Lagrangian neural ODE, using a differentiable Hessian learned by a NN and a total loss to regularize learning. Empirical results on analytical systems (e.g., Kepler, damped oscillator) show that the Helmholtz metrics recover a Hessian close to the true Lagrangian and distinguish non-Lagrangian dynamics, while data-driven experiments demonstrate that Lagrangian-regularized models can match regression performance and improve alignment with EL structure, suggesting practical benefits for physics-informed learning and extrapolation.

Abstract

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.

Paper Structure

This paper contains 8 sections, 6 equations, 3 figures.

Table of Contents

  1. Differential equations in physics and Machine Learning
  2. Related Work
  3. Method: Lagrangian neural ODEs and Helmholtz metrics
  4. Application and evaluation
  5. Helmholtz metrics on analytical ODEs
  6. Lagrangian neural ODEs on data
  7. Conclusion and outlook
  8. Conclusion and outlook

Figures (3)

  • Figure 1: Lagrangian neural ODE model. Gradients are represented by gray arrows.
  • Figure 2: Smoothed loss curves for training Helmholtz metrics on different ODEs for different settings. Left: Oscillator with and without time dependence of $g$. Middle: Kepler problem with different mean eccentricity $\mu_\varepsilon$. Right: non-Lagrangian ODEs. Vertical axes are scaled differently.
  • Figure 3: Results of training a Lagrangian neural ODE on an oscillator.