  We analyse the following inverse problem. Given a nonconvex functional (from
a specific, but quite general class) of normal, codimension-1 currents (which
in two spatial dimensions can be interpreted as transportation networks), find
the potential of a phase field energy which approximates the given functional.
We prove existence of a solution as well as its characterization via a linear
deconvolution problem. We also provide an explicit formula that allows to
approximate the solution arbitrarily well in the supremum norm.
