Neural Symplectic Integrator with Hamiltonian Inductive Bias for the Gravitational $N$-body Problem
This work addresses the challenge of long-term, energy-conserving numerical simulations in astrophysics, offering a novel neural approach that is incremental in combining existing symplectic methods with neural networks.
The authors tackled the gravitational N-body problem by developing a neural symplectic integrator that splits the Hamiltonian into analytically solvable and neural network-approximated parts, achieving accurate integration for 10^5 steps without deviating from ground truth dynamics and showing good generalization to unseen systems.
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.