Lagrangian neural ODEs: Measuring the existence of a Lagrangian with Helmholtz metrics
This addresses the need for physically consistent machine learning models in physics applications, though it appears incremental as it builds on existing neural ODE frameworks.
The authors tackled the problem of ensuring neural ODE solutions are physical by introducing Helmholtz metrics to measure if an ODE resembles an Euler-Lagrange equation, and they developed Lagrangian neural ODEs that learn such equations directly, improving solutions with zero additional inference cost.
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.