Finding symmetry breaking order parameters with Euclidean neural networks

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 as 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.