Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery

TL;DR

This paper introduces a framework for learning discrete Lagrangians and their symmetries directly from motion data, enabling the discovery of conserved quantities without prior assumptions on the Lagrangian's form.

Contribution

It presents a novel method to learn discrete Lagrangians and symmetries from data, leveraging Lie group theory, without requiring velocity data or restrictive assumptions.

Findings

Improves qualitative and quantitative modeling of dynamical systems

Successfully identifies symmetries and conserved quantities from noisy data

Enhances model robustness and accuracy in simulations

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.