Riemannian Liquid Spatio-Temporal Graph Network

2026-01-20

AI Summary

This work addresses the limitation of existing Liquid Time-Constant Networks, which are confined to Euclidean space and struggle to effectively model spatiotemporal graph dynamics with non-Euclidean geometric structures—such as hierarchical or cyclic topologies. The paper presents the first extension of liquid neural networks to Riemannian manifolds, formulating continuous-time ordinary differential equations on curved spaces to incorporate geometric inductive biases that better capture graph evolution. The proposed method offers theoretical stability guarantees and quantifies model expressivity through trajectory-based state analysis. Empirical evaluations demonstrate that the model significantly outperforms current approaches across multiple real-world spatiotemporal graph benchmarks, with particularly pronounced gains in scenarios involving complex non-Euclidean structures.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• 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
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Liquid Time-Constant networks (LTCs), a type of continuous-time graph neural network, excel at modeling irregularly-sampled dynamics but are fundamentally confined to Euclidean space. This limitation introduces significant geometric distortion when representing real-world graphs with inherent non-Euclidean structures (e.g., hierarchies and cycles), degrading representation quality. To overcome this limitation, we introduce the Riemannian Liquid Spatio-Temporal Graph Network (RLSTG), a framework that unifies continuous-time liquid dynamics with the geometric inductive biases of Riemannian manifolds. RLSTG models graph evolution through an Ordinary Differential Equation (ODE) formulated directly on a curved manifold, enabling it to faithfully capture the intrinsic geometry of both structurally static and dynamic spatio-temporal graphs. Moreover, we provide rigorous theoretical guarantees for RLSTG, extending stability theorems of LTCs to the Riemannian domain and quantifying its expressive power via state trajectory analysis. Extensive experiments on real-world benchmarks demonstrate that, by combining advanced temporal dynamics with a Riemannian spatial representation, RLSTG achieves superior performance on graphs with complex structures. Project Page: https://rlstg.github.io

Problem

• Riemannian geometry
• graph neural networks
• non-Euclidean structures
• spatio-temporal graphs
• geometric distortion

Innovation

• Riemannian manifold
• Liquid Time-Constant networks
• spatio-temporal graph
• continuous-time GNN
• geometric deep learning