# Choosing neural network output for prediction (regression) of a dynamical system

I’m trying to train a neural network to approximate the output of a dynamical system $$dy/dt=f\left(y(t), u(t) \right)$$, namely, given $$y(0)$$ and $$u(t_i), i=1,2...N$$ I want the network to predict $$y(t_i), i=1,2...N$$. So far I’ve thought of several approaches, namely

1. Predict the derivative $$dy/dt (t_{i+1}) = f_1 \left(y(t_i), u(t_i) \right)$$ and then compute $$y(t_{i+1}) = dy/dt (t_{i+1}) \cdot dt + y(t_{i})$$

2. Predict the increment $$\Delta y (t_{i+1})= f_2 \left(y(t_i), u(t_i), \Delta t \right)$$ and then compute $$y(t_{i+1}) = \Delta y (t_{i+1}) + y(t_{i})$$

3. Directly predict the next value $$y(t_{i+1}) = f_3 \left(y(t_i), u(t_i), \Delta t \right)$$

Which option is recommended?