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.

Figure 1

Left: The predicted displacement signal for a network trained to deform the rectangle into the square. Right: The predicted displacement signal for the network trained to deform the square into the rectangle. Solid lines outline the input shape and dashed lines outline the desired shape. Arrows point from input geometry to desired geometry.

Figure 2

Input parameters (top row) and output signal (bottom row) for one of the square vertices at (from left to right) the start, middle, and end of the model and order parameter optimization. For simplicity, we only plot components with the same parity as the spherical harmonics ($2_e^2$ and $4_e^2$) as they require less familiarity with how parity behaves to interpret. The starting input parameter (on the left) is only a scalar of value 1 ($0_e^0=1$), hence it being a spherically symmetric signal. As the optimization procedure continues, the symmetry breaking parameters become larger, gaining contribution from components other than $0_e^0$ and the model starts to be able to fit to the target output. When the loss converges, the input parameters have nonzero $2_e^2, 3_e^{-2}, 4_e^2$, and $5_e^{-2}$ components with other nonscalar components close to zero and the model is able to fit to the target output.

Figure 3

Perovskite crystal structure with chemical formula of the form $AB{X}_3$ and parent symmetry of $Pm\overline{3}m$ (221). Octahedra can tilt in alternating patterns. This increases the size of the unit cell needed to describe the crystal structure. The larger unit cell directions are given in terms of the parent unit cell directions. The tilting of rotation axes for the $Pnma$ (62) structure is made of a 3D “checkerboard” of alternating rotations in the plane perpendicular to $a+b$ and a 2D checkerboard of alternating rotations in the $ab$ plane along $c$. The tilting of rotation axes for the $Imma$ (74) structure does not possess any alternating tilting in the $ab$ plane direction around the $c$ direction. The structure in space group $Pnma$ (62) corresponds to Glazer notation $a^{+}{b}^{-}{b}^{-}$ and Ref. [31] notation $\left(a000bb\right)$. The structure in space group $Imma$ (74) corresponds to Glazer notation $a^0{b}^{-}{b}^{-}$ and Ref. [31] notation $\left(0000bb\right)$.