让KAN网络具备任意矩阵群等变性，提升科学建模精度与效率
Incorporating Arbitrary Matrix Group Equivariance into KANs
- 用门控样条基函数与等变线性权重构建新层，实现任意矩阵群等变
- 在粒子散射和三体问题中，测试MSE降低数个数量级，参数更少
- 适合对称性关键的科学计算场景，尤其小数据或低参数需求任务
Kolmogorov-Arnold网络（KAN）因样条激活函数在科学领域表现优异，成为多层感知机（MLP）的替代方案。但样条函数可能不尊重任务中的对称性，而对称性是机器学习中的关键先验知识。本文提出等变KAN（EKAN），可将任意矩阵群等变性融入KAN中，拓展其应用范围。我们首先构建门控样条基函数，与等变线性权重共同组成EKAN层，并设计提升层以对齐输入空间与数据特征空间，从而构建完整架构。相较于基线模型，EKAN在具有对称性的任务（如粒子散射、三体问题）中，使用更小数据集或更少参数即可达到更高精度，测试均方误差常降低数个数量级。即使在非符号公式场景（如含三个喷注的顶夸克识别），EKAN也仅需不到40%参数量即达到当前先进等变架构的性能，而标准KAN未超越MLP。代码与数据见https://github.com/hulx2002/EKAN。
原文摘要 · Abstract (English)
Kolmogorov-Arnold Networks (KANs) have seen great success in scientific domains thanks to spline activation functions, becoming an alternative to Multi-Layer Perceptrons (MLPs). However, spline functions may not respect symmetry in tasks, which is crucial prior knowledge in machine learning. In this paper, we propose Equivariant Kolmogorov-Arnold Networks (EKAN), a method for incorporating arbitrary matrix group equivariance into KANs, aiming to broaden their applicability to more fields. We first construct gated spline basis functions, which form the EKAN layer together with equivariant linear weights, and then define a lift layer to align the input space of EKAN with the feature space of the dataset, thereby building the entire EKAN architecture. Compared with baseline models, EKAN achieves higher accuracy with smaller datasets or fewer parameters on symmetry-related tasks, such as particle scattering and the three-body problem, often reducing test MSE by several orders of magnitude. Even in non-symbolic formula scenarios, such as top quark tagging with three jet constituents, EKAN achieves comparable results with state-of-the-art equivariant architectures using fewer than 40% of the parameters, while KANs do not outperform MLPs as expected. Code and data are available at https://github.com/hulx2002/EKAN .
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