My problem is the following : I have a number of inputs with the corresponding deterministic outputs. There is no error on either input or output. The link between the two is completely unknown to me. With this information, what kind of mathematical technique should I use to get estimated outputs from new inputs ? Are we talking about machine learning, regression,... ? I am even confused about the mathematical field involved here, as statistics seem to give some tools for this kind of problem, but in my case everything is perfectly deterministic.
Function interpolation is what you are trying to achieve. A standard method is to chose a basis of functions (for example polynomials of your input data, if you do polynomial interpolation) and express your input-output relationship as weighted sums of those functions. If you have $n$ known corresponding points, you can chose a basis of $n$ (independent) functions and find the $n$ weights by solving a linear system of $n$ equations.
The previous answer holds true for a single set of input and output. If you have a series of inputs, corresponding outputs and if you are interested in knowing the required input for a specific output then "machine learning" is the answer. I would start with artificial neural network.