A Neural Network Technique for the Derivation of Runge–Kutta Pairs Adjusted for Scalar Autonomous Problems

Abstract

We consider the scalar autonomous initial value problem as solved by an explicit Runge–Kutta pair of orders 6 and 5. We focus on an efficient family of such pairs, which were studied extensively in previous decades. This family comes with 5 coefficients that one is able to select arbitrarily. We set, as a fitness function, a certain measure, which is evaluated after running the pair in a couple of relevant problems. Thus, we may adjust the coefficients of the pair, minimizing this fitness function using the differential evolution technique. We conclude with a method (i.e. a Runge–Kutta pair) which outperforms other pairs of the same two orders in a variety of scalar autonomous problems.