🤖 AI Summary
This work addresses the problem of automatically discovering symmetries and corresponding conservation laws (e.g., energy, linear momentum, angular momentum) of physical systems directly from noisy, discrete trajectory data—without requiring velocity/momentum measurements or prior knowledge of the governing model structure. Methodologically, it introduces the first end-to-end framework for jointly learning discrete Lagrangian functions and their symmetry groups, integrating discrete variational integration, Lie group theory, and neural network parameterization, alongside a symmetry-driven loss function. It further proposes variational backward error analysis to rigorously unify discrete modeling with continuous physical constraints. Experiments demonstrate that the method significantly improves long-term trajectory prediction accuracy under noise, strictly preserves conservation quantities, and achieves superior qualitative and quantitative performance compared to state-of-the-art baselines.
📝 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.