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 no...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 stateof-the-art methods such as the piecewise Padéapproximation,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.展开更多
基金National Key R&D Program of China:Gravitational Wave Detection Project(No.2021YFC22026,No.2021YFC2202601,No.2021YFC2202603)National Natural Science Foundation of China(No.12172288).
文摘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 stateof-the-art methods such as the piecewise Padéapproximation,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.
