A Test-Time Learning Approach to Reparameterize the Geophysical Inverse
  Problem with a Convolutional Neural Network
Anran Xu Lindsey J. Heagy
Regularization is critical in solving the ill-posed geo-physical inversion
problems. Explicit regularization is often used, but there are opportunities to
explore the implicit regularization effect inherently from a Neural Network
structure. Researchers in Computer Vision (CV) have discovered that the
Convolutional Neural Network (CNN) architecture inherently enforces a
regularization that is advantageous for addressing diverse CV inverse problems,
including de-noising and in-painting. In this study, we examine the
applicability of this implicit regularization to geophysical inversions. The
CNN maps an arbitrary vector to the model space (e.g. log-conductivity on the
simulation mesh). The predicted subsurface model is then fed into a forward
numerical simulation process to generate corresponding predicted measurements.
Subsequently, the objective function value is computed by comparing these
predicted measurements with the observed field measurements. The
backpropagation algorithm is employed to update the trainable parameters of the
CNN during the inversion. Note that the CNN in our proposed method does not
require training before the inversion, rather, the CNN weights are estimated in
the inversion algorithm, hence this is a test-time learning (TTL) approach. The
results demonstrate that the implicit regularization provided by the CNN can be
useful in DC resistivity inversions. We also provide a detailed discussion of
the potential sources of this implicit regularization and some practical guides
for applying the proposed method to other geophysical scenarios. The proposed
approach for reparameterizing the inverse problem can be adapted to other
Tikhonov-style geophysical inversions.