The reconstruction of physical fields from sparse measurements is pivotal in both scientific research and engineering applications. Traditional methods are increasingly supplemented by deep learning models due to their efficacy in extracting features from data. However, except for the low accuracy on complex physical systems, these models often fail to comply with essential physical constraints, such as governing equations and boundary conditions. To overcome this limitation, we introduce a novel data-driven field reconstruction framework, termed the Physics-aligned Schr\"{o}dinger Bridge (PalSB). This framework leverages a diffusion bridge mechanism that is specifically tailored to align with physical constraints. The PalSB approach incorporates a dual-stage training process designed to address both local reconstruction mapping and global physical principles. Additionally, a boundary-aware sampling technique is implemented to ensure adherence to physical boundary conditions. We demonstrate the effectiveness of PalSB through its application to three complex nonlinear systems: cylinder flow from Particle Image Velocimetry experiments, two-dimensional turbulence, and a reaction-diffusion system. The results reveal that PalSB not only achieves higher accuracy but also exhibits enhanced compliance with physical constraints compared to existing methods. This highlights PalSB's capability to generate high-quality representations of intricate physical interactions, showcasing its potential for advancing field reconstruction techniques. The source code can be found at https://github.com/lzy12301/PalSB.

Social linking prediction is one of the most fundamental problems in online social networks and has attracted researchers’ persistent attention. Most of the existing works predict unobserved links using graph neural networks (GNNs) to learn node embeddings upon pair-wise relations. Despite promising results given enough observed links, these models are still challenging to achieve heart-stirring performance when observed links are extremely limited. The main reason is that they only focus on the smoothness of node representations on pair-wise relations. Unfortunately, this assumption may fall when the networks do not have enough observed links to support it. To this end, we go beyond pair-wise relations and propose a new and novel framework using hypergraph neural networks with multi-level hyperedge distillation strategies. To break through the limitations of sparsely observed links, we introduce the hypergraph to uncover higher-level relations, which is exceptionally crucial to deduce unobserved links. A hypergraph allows one edge to connect multiple nodes, making it easier to learn better higher-level relations for link prediction. To overcome the restrictions of manually designed hypergraphs, which is constant in most hypergraph researches, we propose a new method to learn high-quality hyperedges using three novel hyperedges distillation strategies automatically. The generated hyperedges are hierarchical and follow the power-law distribution, which can significantly improve the link prediction performance. To predict unobserved links, we present a novel hypergraph neural networks named HNN. HNN takes the multi-level hypergraphs as input and makes the node embeddings smooth on hyperedges instead of pair-wise links only. Extensive evaluations on four real-world datasets demonstrate our model’s superior performance over state-of-the-art baselines, especially when the observed links are extremely reduced.