Speaker
Description
Neural ODEs enable the automatic discovery of a system's ordinary differential equations (ODEs) using only trajectory measurements. We combine this powerful and widely used machine learning method with the principle of stationary action from theoretical physics to learn only ODEs that are admissible as fundamental physical laws. To this end, we develop Helmholtz metrics, a machine learning architecture that quantifies violations of the Helmholtz conditions arising in the inverse problem of the calculus of variations. These conditions determine whether a Lagrangian exists that yields a given ODE as an Euler-Lagrange equation. We then use this measure as a regularization term for Neural ODEs to obtain Lagrangian Neural ODEs, which recover Euler-Lagrange equations directly from positional data. When no Lagrangian exists for a given system, the models reflect this incomaptibility through their convergence behaviour, indicating inconsistencies in the system's physical description. For systems that admit Lagrangian description, ODEs learned in this way are not only more theoretically grounded, but also yield more physically consistent and robust predictions, particularly in sparse and noisy data regimes.