A hybrid approach for solving the gravitational N-body problem with
  Artificial Neural Networks
Veronica Saz Ulibarrena Philipp Horn Simon Portegies Zwart Elena Sellentin Barry Koren Maxwell X. Cai
Simulating the evolution of the gravitational N-body problem becomes
extremely computationally expensive as N increases since the problem complexity
scales quadratically with the number of bodies. We study the use of Artificial
Neural Networks (ANNs) to replace expensive parts of the integration of
planetary systems. Neural networks that include physical knowledge have grown
in popularity in the last few years, although few attempts have been made to
use them to speed up the simulation of the motion of celestial bodies. We study
the advantages and limitations of using Hamiltonian Neural Networks to replace
computationally expensive parts of the numerical simulation. We compare the
results of the numerical integration of a planetary system with asteroids with
those obtained by a Hamiltonian Neural Network and a conventional Deep Neural
Network, with special attention to understanding the challenges of this
problem. Due to the non-linear nature of the gravitational equations of motion,
errors in the integration propagate. To increase the robustness of a method
that uses neural networks, we propose a hybrid integrator that evaluates the
prediction of the network and replaces it with the numerical solution if
considered inaccurate. Hamiltonian Neural Networks can make predictions that
resemble the behavior of symplectic integrators but are challenging to train
and in our case fail when the inputs differ ~7 orders of magnitude. In
contrast, Deep Neural Networks are easy to train but fail to conserve energy,
leading to fast divergence from the reference solution. The hybrid integrator
designed to include the neural networks increases the reliability of the method
and prevents large energy errors without increasing the computing cost
significantly. For this problem, the use of neural networks results in faster
simulations when the number of asteroids is >70.