  Deep learning techniques have shown significant potential in many
applications through recent years. The achieved results often outperform
traditional techniques. However, the quality of a neural network highly depends
on the used training data. Noisy, insufficient, or biased training data leads
to suboptimal results.
  We present a hybrid method that combines deep learning with iterated graph
Laplacian and show its application in acoustic impedance inversion which is a
routine procedure in seismic explorations. A neural network is used to obtain a
first approximation of the underlying acoustic impedance and construct a graph
Laplacian matrix from this approximation. Afterwards, we use a Tikhonov-like
variational method to solve the impedance inversion problem where the
regularizer is based on the constructed graph Laplacian. The obtained solution
can be shown to be more accurate and stable with respect to noise than the
initial guess obtained by the neural network. This process can be iterated
several times, each time constructing a new graph Laplacian matrix from the
most recent reconstruction. The method converges after only a few iterations
returning a much more accurate reconstruction.
  We demonstrate the potential of our method on two different datasets and
under various levels of noise. We use two different neural networks that have
been introduced in previous works. The experiments show that our approach
improves the reconstruction quality in the presence of noise.
