Data over non-Euclidean manifolds, often discretized as surface meshes, naturally arise in computer graphics and biological and physical systems. In particular, solutions to partial differential equations (PDEs) over manifolds depend critically on the underlying geometry. While graph neural networks have been successfully applied to PDEs, they do not incorporate surface geometry and do not consider local gauge symmetries of the manifold. Alternatively, recent works on gauge equivariant convolutional and attentional architectures on meshes leverage the underlying geometry but underperform in modeling surface PDEs with complex nonlinear dynamics. To address these issues, we introduce a new gauge equivariant architecture using nonlinear message passing. Our novel architecture achieves higher performance than either convolutional or attentional networks on domains with highly complex and nonlinear dynamics. However, similar to the non-mesh case, design trade-offs favor convolutional, attentional, or message passing networks for different tasks; we investigate in which circumstances our message passing method provides the most benefit.

Data over non-Euclidean manifolds, often discretized as surface meshes, naturally arise in computer graphics and biological and physical systems. In particular, solutions to partial differential equations (PDEs) over manifolds depend critically on the underlying geometry. While graph neural networks have been successfully applied to PDEs, they do not incorporate surface geometry and do not consider local gauge symmetries of the manifold. Alternatively, recent works on gauge equivariant convolutional and attentional architectures on meshes leverage the underlying geometry but underperform in modeling surface PDEs with complex nonlinear dynamics. To address these issues, we introduce a new gauge equivariant architecture using nonlinear message passing.

Assessing the potential impact of papers is of great significance to both academia and industry (Wang, Song and Barabási, 2013), especially given the exponential annual growth in the number of papers (Lo, Wang, Neumann, Kinney and Weld, 2020; Chu and Evans, 2021; Xue, He, Liu, Jiang, Zhao and Lu, 2023). As the numerical value of the scientific impact could be difficult to determine, citation count is frequently employed as a rough estimate (Evans and Reimer, 2009; Sinatra, Wang, Deville, Song and Barabási, 2016; Jiang, Koch and Sun, 2021). Actually, the dynamics in citation networks cannot be ignored. For example, the "sleeping beauties" (Van Raan, 2004) phenomenon indicates that the citations of a paper can vary considerably in different time periods. Besides the content quality, the future citations of a paper will be influenced by newly published papers (Funk and Owen-Smith, 2017; Park, Leahey and Funk, 2023). New papers may be successors to older ones, discovering the importance of previous works, thereby drawing more citations for them; or new papers may be competing with older ones, correcting or improving the previous works, thus making them lose potential citations. Therefore, it's imperative to capture dynamics of the citation network to accurately predict the future citations of a target paper. Previous studies within informetrics have primarily concentrated on content information or citation networks of papers.