  Since its formulation by Sir Isaac Newton, the problem of solving the
equations of motion for three bodies under their own gravitational force has
remained practically unsolved. Currently, the solution for a given
initialization can only be found by performing laborious iterative calculations
that have unpredictable and potentially infinite computational cost, due to the
system's chaotic nature. We show that an ensemble of solutions obtained using
an arbitrarily precise numerical integrator can be used to train a deep
artificial neural network (ANN) that, over a bounded time interval, provides
accurate solutions at fixed computational cost and up to 100 million times
faster than a state-of-the-art solver. Our results provide evidence that, for
computationally challenging regions of phase-space, a trained ANN can replace
existing numerical solvers, enabling fast and scalable simulations of many-body
systems to shed light on outstanding phenomena such as the formation of
black-hole binary systems or the origin of the core collapse in dense star
clusters.
