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

翻译：液态时间常数网络（LTCs）作为一类连续时间图神经网络，擅长建模不规则采样的动态过程，但其本质上局限于欧几里得空间。当表示具有固有非欧几里得结构（例如层次结构和循环结构）的真实世界图时，这种限制会引入显著的几何失真，从而降低表示质量。为克服这一局限，我们提出了黎曼流形液态时空图网络（RLSTG），该框架将连续时间液态动力学与黎曼流形的几何归纳偏置相统一。RLSTG通过直接在弯曲流形上构建的常微分方程（ODE）来建模图的演化，使其能够忠实地捕捉结构静态和动态时空图的内在几何特性。此外，我们为RLSTG提供了严格的理论保证，将LTCs的稳定性定理扩展至黎曼流形领域，并通过状态轨迹分析量化了其表达能力。在真实世界基准数据集上进行的大量实验表明，通过将先进的时间动力学与黎曼空间表示相结合，RLSTG在具有复杂结构的图上实现了卓越的性能。项目页面：https://rlstg.github.io