Learning Schwarzschild Geodesic Motion with Lagrangian Neural Networks

Lagrangian Neural Networks (LNNs) learn dynamics from trajectory data by parameterizing a scalar Lagrangian and deriving accelerations through the Euler--Lagrange equation, providing a variational inductive bias that improves physical consistency over purely data-driven models. LNNs have recently been extended to general relativistic settings, leaving Schwarzschild geodesics as an open challenge. Training on complex relativistic systems is hindered by the diversity of the dynamics, as well as numerical instabilities and the difficulty of adequately sampling physically relevant, diverse orbital regimes. We introduce two coordinated improvements: a decoupled position--velocity normalization scheme that independently rescales each coordinate and its time derivatives, reweighting acceleration targets to balance contributions across all channels; and an energy-constrained log-uniformly sampled dataset generation strategy informed by the innermost stable circular orbit (ISCO) structure. With these improvements, our model accurately reproduces perihelion precession and relativistic time dilation, conserves energy and angular momentum to within a few percent even after completing multiple orbits around the central mass and predicts orbital trajectories across eight orbital scenarios in close, medium, and far-field regimes. These results establish that LNNs, augmented with physically motivated normalization and sampling, can reliably learn geodesic dynamics in curved spacetime. Our work also represents the first successful training of LNNs on Schwarzschild geodesics and offers a promising direction for data-driven modeling of complex relativistic systems.