用旋转变换球谐函数实现高效且完整的三维原子系统等变网络
Spin-Weighted Spherical Harmonics Enable Complete and Scalable $\mathrm{E}(3)$-Equivariant Networks
- 基于旋转变换球谐函数构建新张量积，解决现有方法表达不全问题
- 复杂度保持GTP水平（约O(L^6)），精度接近完整CGTP
- 特别适合手性材料和非中心对称结构的模拟，适合分子动力学研究者
E(3)等变网络在三维原子系统建模中具有潜力，但其可扩展性受限于克勒巴施-戈登张量积（CGTP）的O(L^6)复杂度。近期提出的盖恩特张量积（GTP）虽降低复杂度，却无法捕捉反对称路径，导致表达不完整。本文提出SpinGTP，通过将标量函数推广至旋转变换球谐函数（SWSH），利用SWSH的代数性质恢复缺失的反对称相互作用，同时保持GTP的渐近效率。该方法还提供更具表现力的等变基，自然包含张量积的奇偶分量。在Tetris、3BPA、SPICE-MACE-OFF和OC20等多个基准上评估显示，SpinGTP精度接近完整CGTP。尤其在涉及手性材料与非中心对称几何的任务中表现更优。本工作为大规模3D原子系统模拟中的高阶等变性提供了完整、可扩展且数学严谨的路径。
原文摘要 · Abstract (English)
$\mathrm{E}(3)$-equivariant networks are promising for 3D atomistic system modeling, yet their scalability is limited by the $O(L^6)$ complexity of the Clebsch-Gordan Tensor Product (CGTP). The recently proposed Gaunt Tensor Product (GTP) reduces the complexity but is unable to capture the antisymmetric paths, resulting in incomplete expressivity. In this work, we present SpinGTP, an approach to overcome the GTP incompleteness by generalizing from scalar functions to Spin-Weighted Spherical Harmonics (SWSH). By relying on the algebraic properties of SWSH, SpinGTP recovers the missing antisymmetric interactions while maintaining the asymptotic efficiency of GTP. It also allows for a more expressive equivariant basis that naturally accounts for the parity-odd components of tensor products. We evaluate SpinGTP across diverse benchmarks, including Tetris, 3BPA, SPICE-MACE-OFF, and OC20. Our results show that SpinGTP achieves accuracies comparable to full CGTP. Notably, by explicitly capturing antisymmetric paths, SpinGTP exhibits superior performance in tasks involving chiral materials and non-centrosymmetric geometries. This work provides a complete, scalable, and mathematically rigorous path toward high-order equivariance in large-scale 3D atomistic system simulations.
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