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

• URL: http://arxiv.org/abs/2510.06367v2
• Date: Mon, 10 Nov 2025 18:25:10 GMT
• Title: Lagrangian neural ODEs: Measuring the existence of a Lagrangian with Helmholtz metrics
• Authors: Luca Wolf, Tobias Buck, Bjoern Malte Schaefer,
• Abstract summary: We present Helmholtz metrics to quantify the resemblance for a given ODE to an Euler-Lagrange equation.<n>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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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