Equivariant Lagrangian Neural Networks with Noether Guarantees

Learning physical dynamics from data supports simulation and control. Lagrangian Neural Networks (LNNs) learn a scalar Lagrangian in generalized coordinates and derive dynamics through the Euler-Lagrange equations, but standard LNNs do not guarantee spatial symmetries or their conservation laws. We introduce Equivariant Lagrangian Neural Networks (EqLNNs), which lift nonlinear generalized coordinates and velocities to features with known linear transformation laws and use an equivariant network to produce an invariant scalar Lagrangian. We show that enforcing invariance in the lifted representation makes the resulting Lagrangian invariant to the corresponding transformations of the original generalized coordinates. By Noether’s theorem, this guarantees the corresponding conserved quantities when the learned Lagrangian is regular. The construction handles translations, articulated and rigid-body systems, and fixed external fields through their remaining symmetries. Across five systems, EqLNN improves over a lifted LNN on four systems, including $2.4$ to $52.5$ times lower median trajectory MAE on the free-body systems. Momentum components tied to enforced symmetries drift by at most $5.7\times 10^{-4}$, while matched lifted LNNs show order-one drift in the corresponding angular-momentum components. EqLNN can improve prediction while making the appropriate conservation laws part of the model rather than something inferred from data.