Transformer with Koopman-Enhanced Graph Convolutional Network for Spatiotemporal Dynamics Forecasting

2025-07-04

AI Summary

Addressing spatiotemporal dynamic forecasting on irregular geometric domains, this paper proposes TK-GCN—a unified framework for jointly modeling complex spatial dependencies and nonlinear temporal evolution. Methodologically, it innovatively integrates Koopman operator theory into a graph convolutional network (K-GCN), enabling an interpretable, approximately linear embedding of high-dimensional nonlinear dynamics while preserving temporal consistency. Subsequently, a Transformer module is introduced to capture long-range temporal dependencies within this latent linear space. This two-stage architecture ensures both geometry-aware spatial encoding and robust long-horizon temporal modeling. Evaluated on cardiac dynamics prediction, TK-GCN achieves statistically significant improvements over state-of-the-art baselines in multi-step forecasting, demonstrating its superior capacity to represent and generalize complex spatiotemporal patterns on irregular domains.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Spatiotemporal dynamics forecasting is inherently challenging, particularly in systems defined over irregular geometric domains, due to the need to jointly capture complex spatial correlations and nonlinear temporal dynamics. To tackle these challenges, we propose TK-GCN, a two-stage framework that integrates geometry-aware spatial encoding with long-range temporal modeling. In the first stage, a Koopman-enhanced Graph Convolutional Network (K-GCN) is developed to embed the high-dimensional dynamics distributed on spatially irregular domains into a latent space where the evolution of system states is approximately linear. By leveraging Koopman operator theory, this stage enhances the temporal consistency during the latent learning. In the second stage, a Transformer module is employed to model the temporal progression within the Koopman-encoded latent space. Through the self-attention mechanism, the Transformer captures long-range temporal dependencies, enabling accurate forecasting over extended horizons. We evaluate TK-GCN in spatiotemporal cardiac dynamics forecasting and benchmark its performance against several state-of-the-art baselines. Experimental results and ablation studies show that TK-GCN consistently delivers superior predictive accuracy across a range of forecast horizons, demonstrating its capability to effectively model complex spatial structures and nonlinear temporal dynamics.

Problem

• Forecasting spatiotemporal dynamics on irregular geometric domains
• Capturing complex spatial correlations and nonlinear temporal dynamics
• Improving predictive accuracy in long-range temporal modeling

Innovation

• Koopman-enhanced GCN for linear latent space
• Transformer for long-range temporal dependencies
• Two-stage framework for spatiotemporal forecasting