Multivariate Time Series Forecasting with Hybrid Euclidean-SPD Manifold Graph Neural Networks

Multivariate Time Series (MTS) forecasting plays a vital role in various real-world applications, such as traffic management and predictive maintenance. Existing approaches typically model MTS data in either Euclidean or Riemannian space, limiting their ability to capture the diverse geometric structures and complex spatio-temporal dependencies inherent in real-world data. To overcome this limitation, we propose the Hybrid Symmetric Positive-Definite Manifold Graph Neural Network (HSMGNN), a novel graph neural network-based model that captures data geometry within a hybrid Euclidean-Riemannian framework. To the best of our knowledge, this is the first work to leverage hybrid geometric representations for MTS forecasting, enabling expressive and comprehensive modeling of geometric properties. Specifically, we introduce a Submanifold-Cross-Segment (SCS) embedding to project input MTS into both Euclidean and Riemannian spaces, thereby capturing spatio-temporal variations across distinct geometric domains. To alleviate the high computational cost of Riemannian distance, we further design an Adaptive-Distance-Bank (ADB) layer with a trainable memory mechanism. Finally, a Fusion Graph Convolutional Network (FGCN) is devised to integrate features from the dual spaces via a learnable fusion operator for accurate prediction. Experiments on three benchmark datasets demonstrate that HSMGNN achieves up to a 13.8 percent improvement over state-of-the-art baselines in forecasting accuracy.