Exact SO(3)-Equivariant Isotropic Kernels for Rotation-Robust Neural Dynamics

Computer Science > Machine Learning arXiv:2610.10626 (cs) [Submitted on 7 Oct 2026] Title: Exact SO(3)-Equivariant Isotropic Kernels for Rotation-Robust Neural Dynamics Authors: Ridham Patel.

计算机科学 > 机器学习 arXiv:2610.10626 (cs) [提交于 2026 年 10 月 7 日] 标题：用于旋转鲁棒神经动力学的精确 SO(3) 等变各向同性核 作者：Ridham Patel。

Abstract: Neural surrogates for vector-valued partial differential equations can fit training data yet change their predictions when the same physical state is expressed in a rotated coordinate frame. We study this failure on three-dimensional Navier—Stokes dynamics observed at irregularly placed points.

摘要：用于向量值偏微分方程的神经代理模型虽然能够拟合训练数据，但当同一物理状态在旋转坐标系中表示时，其预测结果会发生改变。我们针对在不规则分布点上观测到的三维纳维-斯托克斯（Navier—Stokes）动力学研究了这一缺陷。

We introduce the Invariant-Conditioned Isotropic Kernel Neural Operator (IKNO), a compact graph model that builds local interactions from scalar quantities unchanged by rotation and vector directions that rotate with the data. Consequently, rotating the positions and velocities rotates the predicted velocity change in exactly the same way.

我们引入了不变条件各向同性核神经算子（IKNO），这是一种紧凑的图模型，它通过旋转不变的标量和随数据旋转的向量方向来构建局部交互。因此，旋转位置和速度会以完全相同的方式旋转预测的速度变化。

On a held-out test set fixed after model design, training unconstrained graph models on randomly rotated examples reduces but does not eliminate their coordinate dependence. In contrast, IKNO is consistent to numerical precision, matches the forecasting accuracy of a general rotation-aware Tensor Field Network with 5.6 times fewer parameters, and outperforms a parameter-matched graph simulator.

在模型设计后固定的留出测试集上，对随机旋转的样本进行无约束图模型训练可以减少但不能消除其对坐标的依赖性。相比之下，IKNO 在数值精度上保持一致，其预测准确度与通用的旋转感知张量场网络相当，但参数量减少了 5.6 倍，且优于参数匹配的图模拟器。

These results show that a compact, PDE-specialized model can remove coordinate dependence without sacrificing forecasting accuracy.

这些结果表明，一种紧凑的、专门针对偏微分方程的模型可以在不牺牲预测准确性的前提下消除坐标依赖性。