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

arXiv: https://arxiv.org/abs/2606.19303 · HTML: https://mlsanity.com/paper/2606.19303/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 65 references that are in the map.
- First posted on arXiv 17 Jun 2026.

## Builds on (most cited first, 30 of 32)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Enhanced Deep Residual Networks for Single Image Super-Resolution](https://mlsanity.com/paper/1707.02921/) (2017)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Image Super-Resolution Using Very Deep Residual Channel Attention Networks](https://mlsanity.com/paper/1807.02758/) (2018)
- [Image Super-Resolution Using Deep Convolutional Networks](https://mlsanity.com/paper/1501.00092/) (2014)
- [Real-Time Single Image and Video Super-Resolution Using an Efficient Sub-Pixel Convolutional Neural Network](https://mlsanity.com/paper/1609.05158/) (2016)
- [Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting](https://mlsanity.com/paper/1709.04875/) (2017)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Super SloMo: High Quality Estimation of Multiple Intermediate Frames for Video Interpolation](https://mlsanity.com/paper/1712.00080/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels](https://mlsanity.com/paper/1711.08920/) (2017)
- [Frame-Recurrent Video Super-Resolution](https://mlsanity.com/paper/1801.04590/) (2018)
- [Video Frame Interpolation via Adaptive Convolution](https://mlsanity.com/paper/1703.07514/) (2017)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [DiffSTG: Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models](https://mlsanity.com/paper/2301.13629/) (2023)
- [MeshfreeFlowNet: A Physics-Constrained Deep Continuous Space-Time Super-Resolution Framework](https://mlsanity.com/paper/2005.01463/) (2020)
- [Delta-PINNs: physics-informed neural networks on complex geometries](https://mlsanity.com/paper/2209.03984/) (2022)
- [Turbulence Enrichment using Physics-informed Generative Adversarial Networks](https://mlsanity.com/paper/2003.01907/) (2020)
- [PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems](https://mlsanity.com/paper/2410.01337/) (2024)
- [Physics-constrained Deep Learning for Robust Inverse ECG Modeling](https://mlsanity.com/paper/2107.12780/) (2021)
- [Function-on-function kriging, with applications to 3D printing of aortic tissues](https://mlsanity.com/paper/1910.01754/) (2019)
- [Automated Identication of Atrial Fibrillation from Single-lead ECGs Using Multi-branching ResNet](https://mlsanity.com/paper/2306.15096/) (2023)
- [Spatio-Temporal Point Process for Multiple Object Tracking](https://mlsanity.com/paper/2302.02444/) (2023)
- [Geometry-aware Active Learning of Spatiotemporal Dynamic Systems](https://mlsanity.com/paper/2504.19012/) (2025)
- [Transformer with Koopman-Enhanced Graph Convolutional Network for Spatiotemporal Dynamics Forecasting](https://mlsanity.com/paper/2507.03855/) (2025)

## Abstract (arXiv)

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.