Multidimensional Topology-Aware Graph Neural Network Based on Discrete Hodge Theory: Dynamic Adjacency and Deformable Feature Calibration

To address the limitations of traditional graph neural networks (GNNs) in modeling high-order interactions and preserving spatial information for geometry-sensitive tasks, we propose a Multidimensional Topology-Aware Graph Neural Network framework based on discrete Hodge theory. Our approach explicitly models high-order topological relationships among <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex>-cells (nodes, edges, facets) by constructing cell complex structures, while introducing a dynamic adjacency mechanism that integrates spatial proximity and semantic similarity through adaptive thresholding and Transformer-based attention gating. A deformable feature calibration module enhances geometric adaptability by compensating for structural distortions via second-order Taylor expansions. Theoretically, we prove that the orthogonal decomposition guided by the Hodge decomposition theorem— \begin{equation*} \operatorname{Im}\left(B_{k+1}\right) \oplus \operatorname{Im}\left(B_{k}^{T}\right) \oplus \operatorname{Harm}_{k}\tag{1} \end{equation*} ensures stable multiscale feature fusion. A gated message-passing mechanism further balances fairness and performance through trajectory-aware control. Experiments demonstrate significant improvements Hydrogen bond prediction on the QM9 molecular dataset achieves a correlation coefficient of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$0.89(+178 \%$</tex> gain). Instance segmentation on COCO exhibits a 3.8% AP boost in bounding box accuracy. Our continued fraction regularization strategy suppresses gradient explosion, reducing adversarial misclassification rates by 58.2%. This work bridges algebraic topology with geometric deep learning, enabling robust representation of complex systems through hierarchical topological priors and dynamic relational reasoning.