Using neural networks for the derivation of Runge–Kutta–Nyström pairs for integration of orbits

In this paper we present Runge–Kutta–Nyström (RKN) pairs of orders 4(3) and 6(4). We choose a test orbit from the Kepler problem to integrate for a specific tolerance. Then we train the free parameters of the above RKN4(3) and RKN6(4) families to perform optimally. For that we form a neural network approach and minimize its objective function using a differential evolution optimization technique. Finally we observe that the produced pairs outperform standard pairs from the literature for Pleiades orbits and Kepler problem over a wide range of eccentricities and tolerances.

► Runge–Kutta–Nystrom pairs are used for integrating 2-body problem and other orbits.
► We train the coefficients on just one eccentricity, accuracy and interval.
► We get better results for the entire family of orbits.