Classifying the Graph Topology of Non-Hermitian Energy Spectra with Graph Transformer

Xianquan Yan, Hakan Akgün, Kenji Kawaguchi, N. Duane Loh, Ching Hua Lee

Track: Regular Track (Page limit: 6-8 pages)

Keywords: Non-Hermitian Physics, Explainable AI (XAI), Graph Neural Networks, Transformer, Higher-Order Topology, Line Graph, Multigraph Dataset, Inverse Design

Abstract: Classifying non-Hermitian energy spectra under open boundary conditions is an open challenge in physics. This classification is a critical prerequisite for the rational inverse design of systems exhibiting desired dynamics and topological responses. While graph topology has emerged as a promising approach for characterizing these spectra, systematic methods for distilling non-Hermitian spectra into their corresponding graph representations have been lacking. Moreover, the resulting graphs often exhibit complexities that defy manual classification, necessitating machine learning approaches. In this work, we introduce a two-step framework for classifying non-Hermitian spectra based on their graph topologies. The first step employs $\texttt{Poly2Graph}$, an automated, high-performance pipeline that distills non-Hermitian spectra into $\textit{spectral graphs}$ suitable for graph neural networks (GNNs). The second step involves generating a large dataset of these spectral graphs and training a GNN for classification. We propose $\texttt{GnLTransformer}$, a novel architecture featuring dual channels that leverage line graphs to explicitly capture higher-order topological features. $\texttt{GnLTransformer}$ achieves over 99% classification accuracy on our dataset, outperforming standard baselines by 32%. Notably, beyond conventional GNNs, $\texttt{GnLTransformer}$ offers inherent explainability regarding higher-order topology. As a further contribution, we release a new $\textit{multi-graph}$ dataset comprising over 117K spectral graphs.

Submission Number: 13