I am unaware of any effort in that direction, but I did find a 1987 paper by Höllig [1] that seems to do a kind of converse (approximates FEM solutions using splines. I haven't read the paper, but that's my quick take). Recovering splines using a FEM might be an interesting learning exercise, since the analytic solution is known and the model seems simple. The motivating physical model is described by flat splines [2], and Carl de Boor has photographs of physical splines in action [3].