Noether's razor: Learning Conserved Quantities

TL;DR

This paper introduces a method leveraging Noether's theorem to learn conserved quantities and symmetries directly from data, enhancing the modeling of dynamical systems with automatic regularization.

Contribution

It proposes a novel approach to parameterize and learn conserved quantities and symmetries from data using a Bayesian framework, avoiding manual regularization.

Findings

Successfully identifies conserved quantities in harmonic oscillators and n-body systems.

Accurately recovers symmetry groups like U(n) and SE(n).

Improves predictive accuracy and model performance.

Abstract

Symmetries have proven useful in machine learning models, improving generalisation and overall performance. At the same time, recent advancements in learning dynamical systems rely on modelling the underlying Hamiltonian to guarantee the conservation of energy. These approaches can be connected via a seminal result in mathematical physics: Noether's theorem, which states that symmetries in a dynamical system correspond to conserved quantities. This work uses Noether's theorem to parameterise symmetries as learnable conserved quantities. We then allow conserved quantities and associated symmetries to be learned directly from train data through approximate Bayesian model selection, jointly with the regular training procedure. As training objective, we derive a variational lower bound to the marginal likelihood. The objective automatically embodies an Occam's Razor effect that avoids collapse of conservation laws to the trivial constant, without the need to manually add and tune additional regularisers. We demonstrate a proof-of-principle on $n$-harmonic oscillators and $n$-body systems. We find that our method correctly identifies the correct conserved quantities and U($n$) and SE($n$) symmetry groups, improving overall performance and predictive accuracy on test data.