Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery

TL;DR

The paper addresses learning dynamical systems with inherent symmetries by proposing Discrete Lagrangian Neural Networks (DLNN) and an enhanced Symmetric DLNN (SymDLNN). It learns a discrete inverse modified Lagrangian from configuration data, uses a variational integrator for forward simulation, and employs variational backward error analysis to recover a continuous Lagrangian, all while enforcing degeneracy-avoidance and symmetry-based regularization. A symmetry-learning framework automatically identifies affine-linear subgroups via momentum maps, yielding conserved quantities and improved generalization. Experiments on pendulum-on-a-cart and Kepler systems show DLNN outperforms LNN in trajectory recreation and prediction, while SymDLNN jointly discovers symmetries and preserves energy and invariants even under noise, demonstrating practical benefits for long-term structure-preserving dynamics. These methods advance data-driven dynamical modeling by integrating geometric structure, backward error analysis, and automatic symmetry discovery.

Abstract

By one of the most fundamental principles in physics, a dynamical system will exhibit those motions which extremise an action functional. This leads to the formation of the Euler-Lagrange equations, which serve as a model of how the system will behave in time. If the dynamics exhibit additional symmetries, then the motion fulfils additional conservation laws, such as conservation of energy (time invariance), momentum (translation invariance), or angular momentum (rotational invariance). To learn a system representation, one could learn the discrete Euler-Lagrange equations, or alternatively, learn the discrete Lagrangian function $\mathcal{L}_d$ which defines them. Based on ideas from Lie group theory, in this work we introduce a framework to learn a discrete Lagrangian along with its symmetry group from discrete observations of motions and, therefore, identify conserved quantities. The learning process does not restrict the form of the Lagrangian, does not require velocity or momentum observations or predictions and incorporates a cost term which safeguards against unwanted solutions and against potential numerical issues in forward simulations. The learnt discrete quantities are related to their continuous analogues using variational backward error analysis and numerical results demonstrate the improvement such models can have both qualitatively and quantitatively even in the presence of noise.

Figures (7)

• Figure 1: Diagramatic explanation of the modified and inverse modified Lagrangian functions (contin.-continuous, EOM- equations of motion).
• Figure 2: Pendulum on the cart example: Recreation and prediction of the trajectory based on single trajectory observation.
• Figure 3: Kepler example: Recreation and prediction of the trajectory based on single trajectory observation.
• Figure 4: Pendulum on a cart example: Symmetry and energy error test based on single trajectory observation (RP- recreation phase, PP- prediction phase).
• Figure 5: Kepler example: Symmetry and energy error test based on single trajectory observation (RP- recreation phase, PP- prediction phase).