发表机构
Universidad Carlos III de Madrid; Instituto de Ciencias Matemáticas ICMAT (CSIC-UAM-UC3M-UCM)（马德里卡洛斯三世大学; 数学科学研究所（ICMAT））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出群胚等变神经网络理论，用于有界域上的可转向CNN，通过二分等变核定理和输运约束实现局部对称性，实验表明精度提升至少一个数量级。
AI 中文摘要
等变卷积神经网络通常由全局作用于信号空间的群构建。这一假设对于许多有界或分层域是不合适的：环境刚体运动可能仅在域的一部分上可容许，且边界引入了对传递群作用不可见的几何类型。我们发展了群胚等变神经网络理论，其中对称性数据由群胚、选定的局部二分伪群、测度以及输入和输出表示丛组成。对于对象空间上的积分通道，我们证明了二分等变核定理：等变性等价于两点核上的输运约束，其解由每对轨道上的一个联合稳定子交织算子分类。作为案例研究，我们将该理论应用于有界平面域。所得架构通过离线零空间基和稀疏收集-变换-散射操作实现。单独使用泊松-狄利克雷核研究来评估边界感知的归纳偏置；精确逆被证明保持矩形的全局对称性，但不保持一般适当的局部二分。数值结果表明，当对称性无法通过群作用全局实现时，所提出的架构提供了显著优势，并且相对于测试的模型，精度至少提高一个数量级。
英文摘要
Equivariant convolutional neural networks are usually built from a group acting globally on the space of signals. This hypothesis is inappropriate for many bounded or stratified domains: an ambient rigid motion may be admissible only on part of the domain, and the boundary introduces geometric types that are invisible to a transitive group action. We develop a theory of groupoid-equivariant neural networks in which the symmetry datum consists of a groupoid, a selected pseudogroup of local bisections, a measure, and input and output representation bundles. For integral channels on the object space, we prove a bisection-equivariant kernel theorem: equivariance is equivalent to a transport constraint on the two-point kernel, and its solutions are classified by one joint-stabilizer intertwiner on each orbit of pairs. As a case study we apply the theory to bounded planar domains. The resulting architecture is implemented through offline nullspace bases and sparse gather--transform--scatter operations. A Poisson--Dirichlet kernel study is used separately to assess boundary-aware inductive bias; the exact inverse is shown to preserve the global symmetries of the rectangle but not general proper local bisections. The numerical results show that the proposed architectures provide significant advantages when symmetries cannot be globally implemented by group actions and provide an accuracy improvement of at least one order of magnitude with respect to the models tested.
