In machine learning regression is usually used to determine the mapping $f(x)$ when you know some values $y$. However I was wondering if some research is already done on determining a functional. Essentially having a function as input, instead of a vector.
Some searching on the web didn't really show me some progress on that subject, but maybe I am not using the right terminology since I am not a ML expert.
Maybe someone has knowledge of this kind of research?
In general you know or you chose a family of allowed functional $V$ among which you seek that depends on some parameters, for example the coefficient we try to estimate in the linear regression.
In this scenario, given a map $V=V_\vartheta$ depending on some parameters $\vartheta$, a set of observations $\rho_1,\dots,\rho_n$ and the evaluations $v_i=V_\vartheta (\rho_i)\in \mathbb C$, we want to determine the $\hat\vartheta$ that better fits the training set (our observations). This might be attained in several ways (depending on the particular statement of the problem or the tools you have available,...), but usually the problem is stated as the minimum problem: $$\hat\vartheta =\underset{\vartheta \in \Theta}{\text{argmin}} \left\lVert\sum_{i=1}^n V_\vartheta(\rho_i) - v_i \right\rVert$$ where the norm is to be chosen according to the purpose (for example the $\ell_2$-norm) and $\Theta$ is the set of feasible $\vartheta$ for the problem (for example $\mathbb R^m$).
Additional Example
Consider the problem of finding the best numerical quadrature of a function on an interval $\Omega$. Then you are aiming to obtain the functional $I=\int_\Omega$ that is the an functional $I:L^1(\Omega)\rightarrow \mathbb R$.
Here we can chose to find the best approximation of $I$ over a particular family of functionals dependent on some parameters in which our search becomes more feasible. The family is actual the model that we are confident that fits the setup. Some models for machine learning are linear regression, logistic regression, softmax, CART, ...
For this particular problem we can chose the family of functional that evaluates the function (we want to integrate) in two points $x_1$, $x_0$ (the parameters of the family) and estimating the integral $$I\approx V_{(x_1,x_2)}(f)=(f(x_1)+f(x_2))\frac{|\Omega|}{2}.$$
This is merely a possible adaption of a problem that fits your requirements to the framework I proposed and it might help you to erase your doubts.