Neural networks-based solution of the two-body problem

Abstract

This paper presents a novel machine learning approach designed to efficiently solve the classical two-body problem. The inherent structure of the two-body problem involves the integration of a system of second-order nonlinear ordinary differential equations. Conventional numerical integration techniques that rely on small computation steps result in a prolonged computational time. Moreover, calculus has limitations in resolving the two-body problem, inevitably converging towards an unresolved Kepler equation of a transcendental nature. To address this issue, we integrate the conventional analytical solution based on true anomaly with a deep neural network representation of the Kepler equation. This results in a highly accurate closed-form solution that is solely dependent on time, which is termed a learning-based solution to the two-body problem. To enhance the precision, a correction module based on Halley iteration is introduced, which substantially improves the final solution in terms of precision and computational cost. Compared to state-of-the-art methods such as the piecewise Padé approximation, Adomian decomposition method, and modified Mikkola’s method, our approach achieves a computational speedup of several thousand to tens of thousands, while maintaining accuracy in large-scale orbit propagation scenarios. Empirical validation under simulated conditions underscores its effectiveness and potential value for long-term orbit determination.