Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks

TL;DR

This paper introduces a neural network-based approach to automatically generate optimal coefficients for explicit Runge-Kutta methods, demonstrated on the two-body problem, showing improved efficiency over existing methods.

Contribution

It presents a novel methodology using neural networks to design tailored numerical methods for specific differential equations, enhancing solution efficiency.

Findings

The neural network successfully optimized Runge-Kutta coefficients for the two-body problem.

The new method outperformed traditional Runge-Kutta methods in efficiency.

The approach is adaptable to other differential equations.

Abstract

A methodology that can generate the optimal coefficients of a numerical method with the use of an artificial neural network is presented in this work. The network can be designed to produce a finite difference algorithm that solves a specific system of ordinary differential equations numerically. The case we are examining here concerns an explicit two-stage Runge-Kutta method for the numerical solution of the two-body problem. Following the implementation of the network, the latter is trained to obtain the optimal values for the coefficients of the Runge-Kutta method. The comparison of the new method to others that are well known in the literature proves its efficiency and demonstrates the capability of the network to provide efficient algorithms for specific problems.