i seriously doubt (although i haven't proved) that there is a unique production function for a given cost function.
intuition: ignoring corner solutions to the cost minimisation problem, the minimum cost function is the set of poins where isoquant curve has the same slope as the isocost curve. It doesn't contain a full description of any other points of the production function, all you know is that they dont satisfy the slope condition above.
Edit: in fact there is a simple counter example: consider a linear production function where the slope is great enough to induce a corner solution, increasing the slope further will retain the corner solution and hence the minimum cost function. Since more than one production function gives the same cost function, no algorithm can recover the production function from the cost function in the general case.