Let us illustrate how we could train an RNN using data from the solution of a well-known differential equation, namely Newton's equation for oscillatory motion for an object being forced into harmonic oscillations by an applied external force.
We will start with the basic algorithm for solving this type of equations using the Runge-Kutta-4 approach. The first code example is a standalone differential equation solver. It yields positions and velocities as function of time, starting with an initial time \( t_0 \) and ending with a final time.
The data the program produces will in turn be used to train an RNN for a selected number of training data. With a trained RNN, we will then use the network to make predictions for data not included in the training. That is, we will train a model which should be able to reproduce velocities and positions not included in training data.