Scalable Graph Networks for Particle Simulations

TL;DR

This paper introduces a hierarchical graph approach for particle simulations that reduces computational complexity from quadratic to linear, enabling large-scale simulations with high accuracy and better generalization.

Contribution

The authors propose a hierarchical graph method that transforms fully-connected interaction graphs into sparse ones, significantly improving scalability and efficiency for particle system modeling.

Findings

Achieves linear time and space complexity for large particle systems.

Maintains high accuracy and energy conservation in large-scale simulations.

Improves generalization to unseen particle counts.

Abstract

Learning system dynamics directly from observations is a promising direction in machine learning due to its potential to significantly enhance our ability to understand physical systems. However, the dynamics of many real-world systems are challenging to learn due to the presence of nonlinear potentials and a number of interactions that scales quadratically with the number of particles $N$, as in the case of the N-body problem. In this work, we introduce an approach that transforms a fully-connected interaction graph into a hierarchical one which reduces the number of edges to $O(N)$. This results in linear time and space complexity while the pre-computation of the hierarchical graph requires $O(N\log (N))$ time and $O(N)$ space. Using our approach, we are able to train models on much larger particle counts, even on a single GPU. We evaluate how the phase space position accuracy and energy conservation depend on the number of simulated particles. Our approach retains high accuracy and efficiency even on large-scale gravitational N-body simulations which are impossible to run on a single machine if a fully-connected graph is used. Similar results are also observed when simulating Coulomb interactions. Furthermore, we make several important observations regarding the performance of this new hierarchical model, including: i) its accuracy tends to improve with the number of particles in the simulation and ii) its generalisation to unseen particle counts is also much better than for models that use all $O(N^2)$ interactions.