用对称神经网络预测液晶有序参数，提升精度与泛化能力。
On the Equivariant Learning of the $Q$-tensor Order Parameter
- 构建七种循环群对称的神经网络，匹配不同旋转对称性。
- 在未见缺陷构型上误差更低，群阶越大性能越好。
- 适合研究液晶、材料模拟中的对称学习任务。
我们构建并评估了用于从合成微观纹理中预测二维$Q$-张量有序参数的群等变神经网络。基于循环群$C_k$（$k=4,8,16,32,64,128,256$）构造了七种架构，结合权重共享、等变激活函数和正则化技术。通过构造作用于行向量化图像的旋转类置换矩阵群$ ho_{C_k}(g)$，近似实现方形图像上圆域的$ rac{2 heta}{k}$旋转。结果显示，所有七种等变模型均满足$Q$-张量等变约束，精度达到单精度浮点水平。与参数匹配的非等变基准模型对比，无论是否使用数据增强，等变模型始终表现更优，且在未见缺陷配置上泛化更强。性能随群阶增加而提升，表明引入更精细的旋转对称性可降低误差。
原文摘要 · Abstract (English)
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.
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