Towards Complex Dynamic Physics System Simulation with Graph Neural ODEs

The great learning ability of deep learning models facilitates us to comprehend the real physical world, making learning to simulate complicated particle systems a promising endeavour. However, the complex laws of the physical world pose significant challenges to the learning based simulations, such as the varying spatial dependencies between interacting particles and varying temporal dependencies between particle system states in different time stamps, which dominate particles' interacting behaviour and the physical systems' evolution patterns. Existing learning based simulation methods fail to fully account for the complexities, making them unable to yield satisfactory simulations. To better comprehend the complex physical laws, this paper proposes a novel learning based simulation model- Graph Networks with Spatial-Temporal neural Ordinary Equations (GNSTODE)- that characterizes the varying spatial and temporal dependencies in particle systems using a united end-to-end framework. Through training with real-world particle-particle interaction observations, GNSTODE is able to simulate any possible particle systems with high precisions. We empirically evaluate GNSTODE's simulation performance on two real-world particle systems, Gravity and Coulomb, with varying levels of spatial and temporal dependencies. The results show that the proposed GNSTODE yields significantly better simulations than state-of-the-art learning based simulation methods, which proves that GNSTODE can serve as an effective solution to particle simulations in real-world application.

How Expressive are Spectral-Temporal Graph Neural Networks for Time Series Forecasting?

Spectral-temporal graph neural network is a promising abstraction underlying most time series forecasting models that are based on graph neural networks (GNNs). However, more is needed to know about the underpinnings of this branch of methods. In this paper, we establish a theoretical framework that unravels the expressive power of spectral-temporal GNNs. Our results show that linear spectral-temporal GNNs are universal under mild assumptions, and their expressive power is bounded by our extended first-order Weisfeiler-Leman algorithm on discrete-time dynamic graphs. To make our findings useful in practice on valid instantiations, we discuss related constraints in detail and outline a theoretical blueprint for designing spatial and temporal modules in spectral domains. Building on these insights and to demonstrate how powerful spectral-temporal GNNs are based on our framework, we propose a simple instantiation named Temporal Graph GegenConv (TGC), which significantly outperforms most existing models with only linear components and shows better model efficiency.