Leveraging Hamiltonian neural flow for robust single-cell multi-omics integration: application to Alzheimer's disease.

Alzheimer's disease (AD) progression involves complex molecular interactions across multiple biological layers, yet integrating high-dimensional single-cell multi-omics data remains computationally challenging. While Graph Convolutional Networks (GCNs) effectively model cell-gene interaction topologies, they face three critical limitations: over-smoothing in deep architectures, instability under data perturbations, and lack of mechanistic interpretability-obstacles that impede clinical translation. The Hamiltonian Graph Convolutional Network (HGCN), a physics-inspired framework integrating symplectic dynamics with graph-based learning, is proposed in this study, which incorporates energy-conserving Hamiltonian mechanics to address these limitations through: (1) geometric constraints that prevent over-smoothing, (2) stable gradient propagation via symplectic integration, and (3) interpretable phase space representations of cellular states. To validate the effectiveness of the HGCN model, it was evaluated on three single-cell multi-omics datasets: an AD prefrontal cortex dataset, and peripheral blood benchmarks. Meanwhile, differential analysis emerged as the most effective feature extraction strategy in the evaluated experimental setting through systematic preprocessing comparisons. On the AD composite classification task requiring simultaneous prediction of cell type and disease state, HGCN achieved 92.28% accuracy and 0.9228 F1-score, significantly outperforming baseline GCN (88.59% accuracy, 0.8860 F1-score). Phase space visualization revealed biologically meaningful patterns: Inhibitory neurons exhibited heterogeneous subtype structures, while disease states showed symmetric geometric organization suggesting cell-type-invariant pathological mechanisms. Robustness experiments on citation networks demonstrated superior resilience to both feature and structural perturbations compared to standard GCN, with performance advantages increasing under higher perturbation intensities. These results establish HGCN as a robust, interpretable framework for multi-omics integration in complex disease analysis, with potential applications in precision medicine.