  In this work, we describe a new approach that uses variational
encoder-decoder (VED) networks for efficient goal-oriented uncertainty
quantification for inverse problems. Contrary to standard inverse problems,
these approaches are \emph{goal-oriented} in that the goal is to estimate some
quantities of interest (QoI) that are functions of the solution of an inverse
problem, rather than the solution itself. Moreover, we are interested in
computing uncertainty metrics associated with the QoI, thus utilizing a
Bayesian approach for inverse problems that incorporates the prediction
operator and techniques for exploring the posterior. This may be particularly
challenging, especially for nonlinear, possibly unknown, operators and
nonstandard prior assumptions. We harness recent advances in machine learning,
i.e., VED networks, to describe a data-driven approach to large-scale inverse
problems. This enables a real-time goal-oriented uncertainty quantification for
the QoI. One of the advantages of our approach is that we avoid the need to
solve challenging inversion problems by training a network to approximate the
mapping from observations to QoI. Another main benefit is that we enable
uncertainty quantification for the QoI by leveraging probability distributions
in the latent space. This allows us to efficiently generate QoI samples and
circumvent complicated or even unknown forward models and prediction operators.
Numerical results from medical tomography reconstruction and nonlinear
hydraulic tomography demonstrate the potential and broad applicability of the
approach.
