Lagrangian Neural Networks

Miles Cranmer, Sam Greydanus, Stephan Hoyer, Peter Battaglia, David Spergel, Shirley Ho

Keywords: Physics, Unsupervised Learning, Energy, Representation Learning, Dynamics, Lagrangians, Hamiltonians, Differential Equations, Neural ODEs, Invariants, Physical Prior, Inductive Bias

TL;DR: Learn arbitrary Lagrangians using a neural network to exactly conserve a learned energy.

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.

Community Implementations: [ 3 code implementations](https://www.catalyzex.com/paper/arxiv:2003.04630/code)