Lagrangian Neural Networks

TL;DR

Lagrangian Neural Networks (LNNs) are a novel approach that parameterizes arbitrary Lagrangians with neural networks, enabling energy-conserving modeling of physical systems without requiring canonical coordinates.

Contribution

This paper introduces Lagrangian Neural Networks, a new method that learns Lagrangians directly, allowing for energy conservation and applicability in systems where canonical momenta are unknown.

Findings

Successfully modeled energy conservation in a double pendulum

Demonstrated relativity modeling without canonical coordinates

Applied to graphs and continuous systems like the 1D wave equation

Abstract

Accurate models of the world are built upon notions of its underlying symmetries. In physics, these symmetries correspond to conservation laws, such as for energy and momentum. Yet even though neural network models see increasing use in the physical sciences, they struggle to learn these symmetries. In this paper, we propose Lagrangian Neural Networks (LNNs), which can parameterize arbitrary Lagrangians using neural networks. In contrast to models that learn Hamiltonians, LNNs do not require canonical coordinates, and thus perform well in situations where canonical momenta are unknown or difficult to compute. Unlike previous approaches, our method does not restrict the functional form of learned energies and will produce energy-conserving models for a variety of tasks. We test our approach on a double pendulum and a relativistic particle, demonstrating energy conservation where a baseline approach incurs dissipation and modeling relativity without canonical coordinates where a Hamiltonian approach fails. Finally, we show how this model can be applied to graphs and continuous systems using a Lagrangian Graph Network, and demonstrate it on the 1D wave equation.