{"ID":2859550,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.06367","arxiv_id":"2510.06367","title":"Lagrangian neural ODEs: Measuring the existence of a Lagrangian with Helmholtz metrics","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.","short_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...","url_abs":"https://arxiv.org/abs/2510.06367","url_pdf":"https://arxiv.org/pdf/2510.06367v2","authors":"[\"Luca Wolf\",\"Tobias Buck\",\"Bjoern Malte Schaefer\"]","published":"2025-10-07T18:29:03Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.DS\",\"physics.comp-ph\",\"physics.data-an\"]","methods":"[]","has_code":false}