Differential Euler: Designing a Neural Network approximator to solve the Chaotic Three Body Problem

TL;DR

This paper explores the use of neural networks, specifically a detailed experimental setup and benchmark, to approximate solutions to the chaotic three body problem, aiming to replace traditional numerical integrators.

Contribution

It introduces a comprehensive experimental framework and benchmark for neural network models to solve the chaotic three body problem, addressing previous limitations and providing insights into their generalization capabilities.

Findings

Neural networks can learn to approximate the three body problem within certain accuracy thresholds.

Increasing dataset complexity tests the limits of neural network generalization.

The proposed models show potential to replace numerical integrators in practical scenarios.

Abstract

The three body problem is a special case of the n body problem where one takes the initial positions and velocities of three point masses and attempts to predict their motion over time according to Newtonian laws of motion and universal gravitation. Though analytical solutions have been found for special cases, the general problem remains unsolved; the solutions that do exist are impractical. Fortunately, for many applications, we may not need to solve the problem completely, i.e., predicting with reasonable accuracy for some time steps, may be sufficient. Recently, Breen et al attempted to approximately solve the three body problem using a simple neural network. Although their methods appear to achieve some success in reducing the computational overhead, their model is extremely restricted, applying to a specialized 2D case. The authors do not provide explanations for critical decisions taken in their experimental design, no details on their model or architecture, and nor do they publish their code. Moreover, the model does not generalize well to unseen cases. In this paper, we propose a detailed experimental setup to determine the feasibility of using neural networks to solve the three body problem up to a certain number of time steps. We establish a benchmark on the dataset size and set an accuracy threshold to measure the viability of our results for practical applications. Then, we build our models according to the listed class of NNs using a dataset generated from standard numerical integrators. We gradually increase the complexity of our data set to determine whether NNs can learn a representation of the chaotic three body problem well enough to replace numerical integrators in real life scenarios.