In this paper, we use the fading regularization method to solve a biharmonic inverse problem, represented by the Cauchy problem. Two formulations are studied and implemented numerically using a finite element method (FEM).We present numerical reconstructions of the missing data on the inaccessible part of the boundary from the knowledge of over-prescribed noisy data for both smooth and piecewise smooth two-dimensional geometries. Numerical examples validate the convergence, stability and efficiency of the proposed numerical algorithm, as well as its capability to deblur the noisy data.