Newton vs the machine: solving the chaotic three-body problem using deep neural networks (1910.07291v1)

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 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.

• The paper demonstrates that a deep ANN can replicate classical solver precision while achieving speeds up to 100 million times faster.
• The paper employs a feed-forward ANN with 10 hidden layers and ADAM optimization, trained on precise Brutus data to capture chaotic dynamics.
• The paper addresses energy conservation challenges by integrating a projection layer, reducing error from 10⁻² to 10⁻⁵ for improved physical accuracy.

Solving the Chaotic Three-Body Problem Using Deep Neural Networks

The paper "Newton vs the Machine: Solving the Chaotic Three-Body Problem Using Deep Neural Networks" presents an innovative approach to addressing the notoriously complex three-body problem by employing deep artificial neural networks (ANNs). The three-body problem, articulated by Sir Isaac Newton, involves predicting the motion of three gravitationally interacting bodies. Its chaotic nature poses significant computational challenges due to its sensitivity to initial conditions, leading to variable and potentially infinite computational effort using conventional methods.

Summary and Methodological Approach

The authors propose a novel method where a deep ANN is trained on a dataset generated by a highly precise numerical integrator, Brutus. By training on an ensemble of solutions for various initial conditions, the ANN was capable of providing accurate predictions of the three-body system over a bounded time interval. The ANN model demonstrated the ability to solve the problem with a fixed computational cost, achieving speeds up to 100 million times faster than traditional state-of-the-art numerical solvers such as Brutus.

For this paper, the authors considered a specific scenario where the three bodies had equal mass and zero initial velocity, effectively reducing the problem's dimensionality and simplifying the initial condition space. They utilized a feed-forward ANN with 10 hidden layers, each comprising 128 nodes, optimized utilizing the adaptive moment estimation (ADAM) algorithm. The ANN was able to approximate the dynamical interactions accurately, evidenced by a mean absolute error (MAE) on the order of 10^-1 in validation tests.

Key Results and Findings

1. Performance Efficiency: The ANN was able to match the precision of Brutus' outputs while drastically reducing computational times. This suggests potential applicability to simulate many-body systems, such as the development of black-hole binaries or core collapses in star clusters.
2. Sensitivity to Initial Conditions: The system's chaotic behavior, characterized by sensitive dependence on initial conditions, was preserved. The ANN was able to emulate the expected divergence between trajectories starting from slightly varied initial positions, indicating robustness in handling chaotic dynamics.
3. Energy Conservation: Energy conservation posed a challenge, with prediction errors generally around 10^-2. Incorporating a projection layer to map outputs onto the correct energy surface reduced these errors to about 10^-5, aligning more closely with a physical interpretation of the system.

Implications and Future Directions

The research suggests substantial implications for the application of machine learning techniques in computational astrophysics, particularly in scenarios requiring extensive numerical integration of chaotic systems. The practical utility of integrating ANNs with traditional solvers presents an exciting avenue for further exploration. Such hybrid techniques could enhance simulations of large astrophysical systems by efficiently resolving computational bottlenecks.

For future research, the authors envisage extending the ANN framework to incorporate more generalized initial conditions, relaxing assumptions on symmetry and mass equality, and potentially addressing higher-order chaos problems, such as four- or five-body systems. This could profoundly impact the field by potentially redefining the computational strategies used in simulating complex gravitational interactions.

In conclusion, the paper showcases the viable replaceability of classical numerical solvers by machine learning models for the three-body problem over specific intervals, paving the way for new methods in solving N-body problems efficiently and accurately in the field of astrophysics.