On the Equivariant Learning of the $Q$-tensor Order Parameter

We construct and evaluate group-equivariant neural networks for the prediction of the two-dimensional $Q$-tensor order parameter of nematic liquid crystals from synthetically generated microscopic textures. Seven architectures, equivariant to cyclic groups $C_k$ of order $k$ for $k=4,\,8,\,16,\,32,\,64,\,128,\, 256$, are built using a combination of weight-sharing constraints, equivariant activations and regularization techniques. To do this, we construct rotation-like permutation matrix groups with elements $\varrho_{C_k}(g)$ that act on row-wise vectorized images, thereby approximating a $\frac{2π}{k}$ rotation of the circular subdomain on square images. We show that all seven equivariant models satisfy the $Q$-tensor equivariance constraint to within single-precision floating point accuracy. Comparing against approximate parameter-matched non-equivariant benchmarks, with and without data augmentation, we find that the equivariant models consistently achieve lower errors and generalize more robustly to unseen defect configurations. Performance increases with group order, suggesting that the incorporation of finer rotational symmetry leads to lower errors.