Projectile motion is given by a function $y = -9.81 x^2 + ax + b$ for some parameters $a$ and $b$. I'll simply assume for $x$ values to be distanced by 1, so $x_t = t$. I can then easily generate series $y_0, \dots, y_n$ of arbitrary lengths and parameters $a, b$. For simplicity, we can assume that the values stay within some bound, e.g. from -1000 to 1000, so that activation functions can still be used.

Is it (theoretically) possible to train an RNN such that for an input series $y_0, \dots, y_n$ it will correctly output the value $y_{n+1}$? Or is the quadratic part in the definition of $y$ not something that an RNN can ever compute?

Note that I'm not talking about some approximation on a compact domain but actual precise computation of the target value.


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