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 Schr\"{o}dinger 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.

翻译：从稀疏测量中重建物理场在科学研究和工程应用中至关重要。传统方法正日益被深度学习模型所补充，因为这些模型在从数据中提取特征方面表现出高效性。然而，除了在复杂物理系统上精度较低外，这些模型通常无法满足关键的物理约束，如控制方程和边界条件。为了克服这一局限，我们引入了一种新颖的数据驱动场重建框架，称为物理对齐的薛定谔桥（PalSB）。该框架利用了一种专门为对齐物理约束而设计的扩散薛定谔桥机制。PalSB方法采用了一个双阶段训练过程，旨在同时处理局部重建映射和全局物理原理。此外，还实施了一种边界感知采样技术，以确保遵守物理边界条件。我们通过在三个复杂非线性系统中的应用展示了PalSB的有效性：粒子图像测速实验中的圆柱绕流、二维湍流以及一个反应-扩散系统。结果表明，与现有方法相比，PalSB不仅实现了更高的精度，而且在遵守物理约束方面表现出更强的能力。这突显了PalSB生成复杂物理相互作用高质量表示的能力，展示了其在推进场重建技术方面的潜力。