  Inverse problems often involve matching observational data using a physical
model that takes a large number of parameters as input. These problems tend to
be under-constrained and require regularization to impose additional structure
on the solution in parameter space. A central difficulty in regularization is
turning a complex conceptual model of this additional structure into a
functional mathematical form to be used in the inverse analysis. In this work
we propose a method of regularization involving a machine learning technique
known as a variational autoencoder (VAE). The VAE is trained to map a
low-dimensional set of latent variables with a simple structure to the
high-dimensional parameter space that has a complex structure. We train a VAE
on unconditioned realizations of the parameters for a hydrological inverse
problem. These unconditioned realizations neither rely on the observational
data used to perform the inverse analysis nor require any forward runs of the
physical model, thus making the computational cost of generating the training
data minimal. The central benefit of this approach is that regularization is
then performed on the latent variables from the VAE, which can be regularized
simply. A second benefit of this approach is that the VAE reduces the number of
variables in the optimization problem, thus making gradient-based optimization
more computationally efficient when adjoint methods are unavailable. After
performing regularization and optimization on the latent variables, the VAE
then decodes the problem back to the original parameter space. Our approach
constitutes a novel framework for regularization and optimization, readily
applicable to a wide range of inverse problems. We call the approach RegAE.
