AI 中文总结
针对散度自由算子学习中的冗余表示问题，提出利用傅里叶空间横截结构通过Householder变换构造最小(D-1)分量表示，实现唯一可逆坐标下的时间演化学习，降低预测误差并增强鲁棒性。
AI 中文摘要
散度自由向量场是不可压缩流动和许多偏微分方程系统中的基本状态变量。冗余参数化，包括神经守恒定律（NCL）势，将多个辅助表示映射到同一物理场。我们的实验表明，这种冗余可以通过扩大等价解集来降低静态表示拟合误差，但由此产生的多对一映射并未为算子学习提供唯一状态。我们引入了一种最小表示，将D维域上的实D分量散度自由向量场编码为同一域上的实(D-1)分量场。利用傅里叶空间中不可压缩性所施加的横截结构，我们使用Householder正交变换直接构造约化坐标。对于周期性和封闭不可渗透场，该变换是可逆的、等距的且保角的。对于开放非周期流，傅里叶延拓构造了一个兼容的周期场，最小能量规则选择唯一的约化表示。神经算子随后完全在此约化空间中学习时间演化。在推理时，预测的(D-1)分量场被直接解码为物理散度自由的D分量场，无需预测环境场或应用事后投影。静态拟合和时间预测实验揭示了任务相关的权衡：冗余有利于静态优化，而唯一可逆坐标则为时间动态提供了定义良好的状态。通过从学习状态空间中移除无约束的纵向或零方向，所提出的公式在保持散度自由的同时实现了更低的预测误差和更强的鲁棒性。
英文摘要
Divergence-free vector fields are fundamental state variables in incompressible flows and many PDE systems. Redundant parameterizations, including Neural Conservation Law (NCL) potentials, map multiple auxiliary representations to the same physical field. Our experiments show that this redundancy can reduce static representation-fitting error by enlarging the set of equivalent solutions, but the resulting many-to-one mapping does not provide a unique state for operator learning. We introduce a minimal representation that encodes a real \(D\)-component divergence-free vector field on a \(D\)-dimensional domain as a real \((D-1)\)-component field on the same domain. Exploiting the transverse structure imposed by incompressibility in Fourier space, we use a Householder orthogonal transformation to construct the reduced coordinates directly. For periodic and closed impermeable fields, the transform is invertible, isometric, and angle-preserving. For open nonperiodic flows, Fourier extension constructs a compatible periodic field, and a minimum-energy rule selects a unique reduced representation. Neural operators then learn temporal evolution entirely in this reduced space. At inference, the predicted \((D-1)\)-component field is decoded directly into a physical divergence-free \(D\)-component field, without predicting an ambient field or applying post-hoc projection. Experiments on static fitting and temporal prediction reveal a task-dependent trade-off: redundancy facilitates static optimization, whereas unique invertible coordinates provide a well-defined state for temporal dynamics. By removing unconstrained longitudinal or null directions from the learned state space, the proposed formulation achieves lower prediction error and greater robustness while preserving divergence freedom by construction.
