Newton versus the machine: solving the chaotic three-body problem using deep neural networks

Monthly Notices of the Royal Astronomical Society, vol. 494, pp. 2465–2470

Abstract

ABSTRACT 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 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.

Authors 4

1. Philip G. Breen corresponding

   Maxwell Institute for Mathematical Sciences · University of Edinburgh

   Affiliation as printed

   Roar AI, 3rd Floor, 116 Dundas Street, Edinburgh EH3 5DQ, UK

   School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Kings Buildings, Edinburgh EH9 3JZ, UK

2. Christopher N. Foley corresponding

   University of Cambridge

   Affiliation as printed

   School of Clinical Medicine, University of Cambridge, Cambridge, CB2 0SP, UK

3. Tjarda Boekholt

   University of Aveiro

   Affiliation as printed

   CIDMA, Departamento de Física, Universidade de Aveiro, Campus de Santiago, P-3810-193 Aveiro, Portugal

4. Simon Portegies Zwart Aachen Leiden Observatory

   Leiden University · Leiden Observatory

   Affiliation as printed

   Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, the Netherlands