Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks

TL;DR

This paper shows that Euclidean symmetry-equivariant neural networks can identify symmetry-breaking parameters in physical systems, adhering to Curie's principle, and formulates these as optimization problems with demonstrated numerical examples.

Contribution

It introduces a mathematical proof and practical demonstration that Euclidean neural networks can find order parameters related to symmetry breaking, aligning with physical principles.

Findings

Neural networks uphold Curie's principle

They can identify symmetry-breaking inputs in physical models

Demonstrated on deforming squares and perovskite tilting patterns

Abstract

Curie's principle states that "when effects show certain asymmetry, this asymmetry must be found in the causes that gave rise to them". We demonstrate that symmetry equivariant neural networks uphold Curie's principle and can be used to articulate many symmetry-relevant scientific questions into simple optimization problems. We prove these properties mathematically and demonstrate them numerically by training a Euclidean symmetry equivariant neural network to learn symmetry-breaking input to deform a square into a rectangle and to generate octahedra tilting patterns in perovskites.