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... Show moreSince 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 converged solutions for the planar chaotic three-body problem 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 a fixed computational cost and up to 100 million times faster than the numerical integrator. In addition, we demonstrate the importance of training an ANN using converged solutions from an arbitrary precise integrator, relative to solutions computed by a conventional fixed precision integrator, which can introduce errors in the training data, due to numerical round-off and time discretization, that are learned by the ANN. 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. Show less
Boekholt, T.C.N.; Portegies Zwart S.F.; Valtonen, M. 2020
Ever since Isaac Newton in 1687 posed the N-body problem, astronomers have been looking for its solutions in order to understand the evolution of dynamical systems, such as our own solar system,... Show moreEver since Isaac Newton in 1687 posed the N-body problem, astronomers have been looking for its solutions in order to understand the evolution of dynamical systems, such as our own solar system, star clusters and galaxies. The main difficulty is that small errors grow exponentially, so that numerical solutions diverge easily from the mathematical solution. This thesis presents two new state of the art N-body algorithms, one of which is designed for high precision (Brutus) and the other for speed (Sakura). The assumption that N-body results are accurate in a statistical sense, is put to the test for three-body configurations. Finally, a new mathematical model is constructed that describes the origin of chaos in a dynamical systems, and explains the short Liapounov time of Comet Halley's orbit. Show less
