  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.
