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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?

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