发表机构
Indian Institute of Technology Gandhinagar（印度甘地纳格尔理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对向量值偏微分方程神经代理的坐标依赖问题，提出IKNO模型，实现旋转鲁棒性，参数仅为通用旋转感知张量场网络的1/5.6，且预测精度相当。
AI 中文摘要
针对向量值偏微分方程的神经代理模型虽能拟合训练数据，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。我们研究了在不规则分布点观测到的三维纳维-斯托克斯动力学中的这种失效问题。我们提出了不变量条件各向同性核神经算子（Invariant-Conditioned Isotropic Kernel Neural Operator, IKNO），这是一种紧凑的图模型，它从不受旋转影响的标量量和随数据旋转的向量方向构建局部交互作用。因此，旋转位置和速度会使预测的速度变化以完全相同的方式旋转。在模型设计后固定的保留测试集上，对随机旋转的示例训练无约束图模型会降低但无法消除其坐标依赖性。相比之下，IKNO在数值精度上保持一致，达到了通用旋转感知张量场网络的预测精度，而参数数量仅为其5.6倍，并且优于参数匹配的图模拟器。这些结果表明，紧凑的偏微分方程专用模型可在不牺牲预测精度的情况下消除坐标依赖性。
英文摘要
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. 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. 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. These results show that a compact, PDE-specialized model can remove coordinate dependence without sacrificing forecasting accuracy.
CommentsAccepted at NeurIPS 2026 Workshop NeurReps (Proceedings Track)
