Higher-order couplings enhance the expressive power of hypergraph neural networks (HGNNs), but they also intensify representation collapse in deep propagation due to strong multi-way feature mixing. This work investigates hypergraph oversmoothing from a dynamical-systems perspective and develops a reaction--diffusion framework for depth-resistant hypergraph learning. By defining hypergraph gradient and divergence operators, we interpret message passing as an incidence-level diffusion process. The analysis of pure diffusion shows that its continuous semiflow exponentially contracts the null-mode-free component of node representations and drives the Dirichlet energy to zero, revealing hypergraph oversmoothing as an intrinsic transverse-energy dissipation phenomenon. Motivated by this analysis, we propose Hypergraph Neural Reaction--Diffusion (HNRD), which introduces a reaction mechanism acting on the transverse component to compensate diffusion-induced dissipation and stabilize discriminative variations. We establish global well-posedness of the proposed dynamics and prove that the null-mode-free Dirichlet energy remains bounded away from zero. A forward-Euler discretization provides a practical HNRD layer with a stability condition for deep propagation. Experiments on benchmark and synthetic heterophilic hypergraphs demonstrate that HNRD consistently improves over representative hypergraph baselines. Depth, robustness, and efficiency analyses further show that HNRD preserves stable performance and nonzero Dirichlet energy under deep propagation and perturbations. These results provide a principled dynamical framework for designing deep hypergraph architectures that maintain higher-order expressiveness without representation collapse.
Xiaofeng Xu, Tingting Dan, Zifan Zhou +3cs.LG math-ph
Accurately predicting the spatiotemporal evolution of amyloid-$β$ and tau proteins at the individual level is critical for improving the diagnosis and treatment of Alzheimer's disease. We consider the problem of constructing patient-specific digital twins that model the propagation of these biomarkers on the cortical surface using reaction--diffusion dynamics. A major challenge is that the underlying nonlinear aggregation mechanisms are unknown and must be inferred from sparse, noisy, and heterogeneous longitudinal PET imaging data. To address this, we develop a data-driven framework that learns biomarker dynamics directly from clinical observations. The approach combines operator learning with reduced-order representations to infer governing equations of disease progression from data. Using this framework, we achieve predictive accuracies of 87\% for amyloid-$β$ and 81\% for tau. Building on the learned dynamics, we further formulate a PDE-constrained optimal control problem to design personalized therapeutic strategies that regulate pathological protein propagation. By integrating data-driven dynamical modeling with treatment optimization, the proposed digital twin framework provides an interpretable and predictive platform for understanding disease progression and enabling precision interventions in neurodegenerative disorders.
Video-language models are pivotal for tasks such as moment retrieval and highlight detection, yet they often struggle to capture the dynamic, non-linear interactions between temporal video sequences and textual semantics. Existing approaches, relying on static cross-attention or prompt-tuning mechanisms, fail to adaptively model the evolving relationships between modalities, leading to suboptimal alignment and limited generalization. Inspired by systems biology, we propose \textbf{Reaction-Diffusion Multimodal Fusion (RDMF)}, a novel framework that reimagines video-language alignment as a reaction-diffusion (RD) process, drawing on the principles of pattern formation introduced by Alan Turing. In RDMF, video features diffuse across time to capture temporal context, while text-video interactions are modeled as non-linear reactions that amplify relevant features and suppress noise, forming emergent patterns akin to biological systems. Leveraging the Gray-Scott RD model, we design a computationally efficient fusion module that integrates video and text representations, supported by rigorous mathematical analysis of stability and convergence using Turing instability criteria. Our framework is theoretically grounded, employing advanced mathematical tools to ensure stable pattern formation, and is practically viable, incorporating standard components like pretrained encoders and DETR-style heads for moment retrieval and saliency prediction. RDMF represents a pioneering interdisciplinary approach, bridging systems biology and multimedia research to address the limitations of conventional multimodal fusion. Preliminary experiments demonstrate its potential to outperform existing methods in identifying salient video moments, offering a new paradigm for video-language tasks.