P-K-GCN: Physics-augmented Koopman-enhanced Graph Convolutional Network for Deep Spatiotemporal Super-resolution

2026-06-17

AI Summary

This work addresses the challenge of high-fidelity reconstruction of low-resolution spatiotemporal data on irregular geometries by proposing a physics-informed graph neural network for super-resolution. The method integrates spline-based graph convolution to capture spatial dependencies, employs Koopman operator embedding to linearize nonlinear dynamics, and incorporates a physics-informed loss to enforce physical consistency. Innovatively embedding Koopman theory and physics-aware mechanisms into a graph neural framework, the approach is theoretically shown to reduce Rademacher complexity and tighten generalization bounds. Evaluated on 3D cardiac electrophysiology reconstruction, the model substantially outperforms existing baselines, successfully recovering high-accuracy spatiotemporal dynamics from sparse, low-resolution observations.

Abstract

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.

Problem

• spatiotemporal super-resolution
• irregular geometries
• physical constraints
• nonlinear dynamics
• high-fidelity simulation

Innovation

• Physics-informed learning
• Koopman operator
• Graph Convolutional Network
• Spatiotemporal super-resolution
• Irregular geometries