Incorporating Arbitrary Matrix Group Equivariance into KANs

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](https://github.com/hulx2002/EKAN).

As an emerging model in the field of science, KAN was never taught how to adhere to symmetry. This lack of capability makes it prone to losing direction when information is insufficient—but we will attempt to teach it.First, we will re-stratify KAN—like slicing a cake. Then, we modify each layer individually, instructing it on the rules it should follow, with each performing its own role. Finally, we assemble them into a cohesive whole, ensuring the new architecture systematically obeys order.Symmetry guidance will empower KAN with stronger learning capabilities—enabling it to infer broader patterns from limited examples and achieve greater efficiency.