Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks
This work provides a method for analyzing symmetry breaking in materials science and geometry, though it appears incremental as it applies known equivariant networks to specific problems.
The paper demonstrates that Euclidean symmetry equivariant neural networks can be used to formulate symmetry-related scientific questions as optimization problems, proving this mathematically and showing numerical results for deforming a square into a rectangle and generating octahedra tilting patterns in perovskites.
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.