The isomorphism problem is a key challenge in both graph and hypergraph domains, crucial for applications like protein design, chemical pathways, and community detection.Hypergraph isomorphism, which models high-order relationships in real-world scenarios, remains underexplored compared to the graph isomorphism.Current algorithms for hypergraphs, like the 1-dimensional generalized Weisfeiler-Lehman test (1-GWL), lag behind advancements in graph isomorphism tests, limiting most hypergraph neural networks to 1-GWL's expressive power.To address this, we propose the high-dimensional GWL (k-GWL), generalizing k-WL from graphs to hypergraphs.We prove that k-GWL reduces to k-WL for simple graphs, and thus develop a unified isomorphism method for both graphs and hypergraphs. We also successfully establish a clear and complete understanding of the GWL hierarchy of expressivity, showing that (k+1)-GWL is more expressive than k-GWL with illustrative examples.Based on k-GWL, we develop a hypergraph neural network model named k-HNN with improved expressive power of k-GWL, which achieves superior performance on real-world datasets, including a 6\% accuracy improvement on the Steam-Player dataset over the runner-up.Our code is available at https://github.com/talence-zcq/KGWL.

The expressive power of graph machine learning characterizes what functions the models can approximate, or equivalently how strong they can distinguish two similar but different hypergraphs. We establish a generalized WL hierarchy for hypergraphs with increasing expressivity, with the notable difference between 1-GFWL vs. 2-GOWL, unlike its graph counterpart. The hierarchy allows us to design hypergraph neural networks with the desired expressivity, improves upon most existing hypergraph neural networks whose expressive power is upper bounded by 1-GWL.