Runge-Kutta method and neural networks for numerical approximation of ordinary differential equations

Abstract

In this work, we present an approach that combines traditional numerical methods for solving differential equations with machine learning techniques, aiming to obtain more efficient approximations of solutions to ordinary differential equations (ODEs). In particular, we propose integrating the Runge-Kutta method to solve initial value problems with neural networks. The Runge-Kutta method is initially employed to generate approximations of the solution in a neighborhood of the initial point. Subsequently, we train a neural network using these points as input data. The results obtained demonstrate that the resulting model performs comparably to the neural network trained solely with points from the real solution. In the comparison with the Runge-Kutta method over the entire interval, using the same number of points, the model performed better. However, we observed that the neural network architecture implemented in the modeling process was not able to outperform the Runge-Kutta method over the entire interval, when we use the same step length for obtaining the points. This result suggests the need for further investigations in the design of the neural network architecture to improve its generalization capability in ODE problems.