Learning from derivative data In many machine learning algorithm, it is often assumed that outputs of unknown function and their corresponding inputs are given to estimate the unknown function. However, I wonder whether there exist some algorithms which can estimate unknown function using derivative data of unknown function and their corresponding inputs. For example,

$$
\mbox{Given }D = {(x_1, y_1),\ldots,(x_n, y_n)}$$
$$\mbox{ where }y_{1} = \frac{d{f(x)}}{dx}\Bigg|_{x=x_1},\ldots,y_n=\frac{d{f(x)}}{dx}\Bigg|_{x=x_n}$$
$$\mbox{ Find  } f(x).$$

I found one curve fitting method which is called Hermite interpolation in wiki and some papers about Hermite learning. However, in these methods, outputs of unknown function and derivative outputs of it and corresponding inputs are given. Especailly, I want to know there exist some methods that utilize only derivative data. Also is it possible to find unknown function from derivative data, in particular, the error bound between unknown function and our estimator can decrease as the number of training data increases. How can we prove it?
If your function is $f:\mathbb{R}\to\mathbb{R}$, you can "learn" $f'$ as a standard regression problem (linear, polynomial, etc.) and then recover $f$ up to an additive constant by integrating $f'$. Obviously, you will only able to recover $f$ up to an additive constant from the derivative.

Update: As for error estimates, here's a simple one. Suppose you've managed to recover $f'$ up to $\epsilon$-accuracy in $L_\infty$. Suppose further that you know that $f(0)=0$ and you only care about $f$ on $[0,T]$. Then your estimate of $f$ (obtained by integrating $f'$) is accurate up to $T\epsilon$ in $L_\infty$.