Learning Relativistic Geodesics and Chaotic Dynamics via Stabilized Lagrangian Neural Networks

Lagrangian Neural Networks (LNNs) can learn arbitrary Lagrangians from trajectory data, but their unusual optimization objective leads to significant training instabilities that limit their application to complex systems. We propose several improvements that address these fundamental challenges, namely, a Hessian regularization scheme that penalizes unphysical signatures in the Lagrangian's second derivatives with respect to velocities, preventing the network from learning unstable dynamics, activation functions that are better suited to the problem of learning Lagrangians, and a physics-aware coordinate scaling that improves stability.

We systematically evaluate these techniques alongside previously proposed methods for improving stability. Our improved architecture successfully trains on systems of unprecedented complexity, including triple pendulums, and achieved 96.6% lower validation loss value and 90.68% better stability than baseline LNNs in double pendulum systems.

With the improved framework, we show that our LNNs can learn Lagrangians representing geodesic motion in both non-relativistic and general relativistic settings. To deal with the relativistic setting, we extended our regularization to penalize violations of Lorentzian signatures, which allowed us to predict a geodesic Lagrangian under AdS\textsubscript{4} spacetime metric directly from trajectory data, which to our knowledge has not been done in the literature before.

This opens new possibilities for automated discovery of geometric structures in physics, including extraction of spacetime metric tensor components from geodesic trajectories. While our approach inherits some limitations of the original LNN framework, particularly the requirement for invertible Hessians, it significantly expands the practical applicability of LNNs for scientific discovery tasks.