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Simulating the evolution of the gravitational N-body problem becomesextremely computationally expensive as N increases since the problem complexityscales quadratically with the number of bodies. We study the use of ArtificialNeural Networks (ANNs) to replace expensive parts of the integration ofplanetary systems. Neural networks that include physical knowledge have grownin popularity in the last few years, although few attempts have been made touse them to speed up the simulation of the motion of celestial bodies. We studythe advantages and limitations of using Hamiltonian Neural Networks to replacecomputationally expensive parts of the numerical simulation. We compare theresults of the numerical integration of a planetary system with asteroids withthose obtained by a Hamiltonian Neural Network and a conventional Deep NeuralNetwork, with special attention to understanding the challenges of thisproblem. Due to the non-linear nature of the gravitational equations of motion,errors in the integration propagate. To increase the robustness of a methodthat uses neural networks, we propose a hybrid integrator that evaluates theprediction of the network and replaces it with the numerical solution ifconsidered inaccurate. Hamiltonian Neural Networks can make predictions thatresemble the behavior of symplectic integrators but are challenging to trainand in our case fail when the inputs differ ~7 orders of magnitude. Incontrast, Deep Neural Networks are easy to train but fail to conserve energy,leading to fast divergence from the reference solution. The hybrid integratordesigned to include the neural networks increases the reliability of the methodand prevents large energy errors without increasing the computing costsignificantly. For this problem, the use of neural networks results in fastersimulations when the number of asteroids is >70.

A hybrid approach for solving the gravitational N-body problem with Artificial Neural Networks