这篇论文提出P-K-GCN，用图卷积加Koopman算子做时空超分辨率，在3D心脏建模上比现有方法更准，物理约束让结果更可靠。
本文提出P-K-GCN框架，结合连续样条GCN从粗粒度图提取空间依赖，并引入Koopman算子理论将非线性时间动力学线性化到紧凑潜空间。优化目标加入物理损失，确保重建结果符合物理定律。理论分析证明物理增强和Koopman正则化通过降低Rademacher复杂度收紧泛化界，减小超分辨率误差。在3D心脏几何上从稀疏低分辨率测量重建高分辨率电动力学，P-K-GCN相比基线模型取得更优精度。
P-K-GCN: Physics-augmented Koopman-enhanced Graph Convolutional Network for Deep Spatiotemporal Super-resolution
High-fidelity simulation of spatiotemporal dynamics is computationally prohibitive, necessitating efficient super-resolution techniques to reconstruct high-resolution data from coarse-grained inputs. Traditional data-driven methods often lack physical constraints, and simple physics-informed learning struggles with irregular spatial geometries and intricately evolving temporal dynamics. To tackle these challenges, we propose a Physics-augmented Koopman-enhanced Graph Convolutional Network (P-K-GCN) for spatiotemporal super-resolution on irregular geometries. Specifically, a continuous spline-based GCN is first designed to extract spatial dependencies directly from coarse graph, and Koopman operator theory is incorporated to project the nonlinear dynamics into a compact latent space where temporal progression is linearized. Second, we augment the optimization objective with a physics-based loss to force the data-driven reconstructions to adhere to physical laws for improving predictive fidelity and robustness. Finally, we provide a rigorous theoretical analysis, establishing that the physics augmentation and Koopman regularization mathematically guarantees a reduction in super-resolution error by diminishing Rademacher complexity and tightening generalization bounds. We evaluate our framework on reconstructing spatially high-resolution cardiac electrodynamics across a 3D heart geometry from sparse low-resolution measurements. Numerical experiments demonstrate that our method achieves superior accuracy compared to baseline models.