Neural Symplectic Integrator with Hamiltonian Inductive Bias for the Gravitational $N$-body Problem

TL;DR

This paper introduces a neural symplectic integrator for the gravitational N-body problem that accurately predicts long-term dynamics and conserves physical quantities, leveraging Hamiltonian splitting and inductive bias.

Contribution

It presents a novel neural network-based integrator that incorporates Hamiltonian structure and symplectic splitting, enabling accurate long-term simulation of N-body systems.

Findings

Successfully integrates three-body systems for 10^5 steps without divergence.

Predicts evolution of unseen N-body systems, demonstrating strong inductive bias.

Maintains conservation of energy and angular momentum over long simulations.

Abstract

The gravitational $N$-body problem, which is fundamentally important in astrophysics to predict the motion of $N$ celestial bodies under the mutual gravity of each other, is usually solved numerically because there is no known general analytical solution for $N>2$. Can an $N$-body problem be solved accurately by a neural network (NN)? Can a NN observe long-term conservation of energy and orbital angular momentum? Inspired by Wistom & Holman (1991)'s symplectic map, we present a neural $N$-body integrator for splitting the Hamiltonian into a two-body part, solvable analytically, and an interaction part that we approximate with a NN. Our neural symplectic $N$-body code integrates a general three-body system for $10^{5}$ steps without diverting from the ground truth dynamics obtained from a traditional $N$-body integrator. Moreover, it exhibits good inductive bias by successfully predicting the evolution of $N$-body systems that are no part of the training set.