  Inverse problems are pervasive mathematical methods in inferring knowledge
from observational and experimental data by leveraging simulations and models.
Unlike direct inference methods, inverse problem approaches typically require
many forward model solves usually governed by Partial Differential Equations
(PDEs). This a crucial bottleneck in determining the feasibility of such
methods. While machine learning (ML) methods, such as deep neural networks
(DNNs), can be employed to learn nonlinear forward models, designing a network
architecture that preserves accuracy while generalizing to new parameter
regimes is a daunting task. Furthermore, due to the computation-expensive
nature of forward models, state-of-the-art black-box ML methods would require
an unrealistic amount of work in order to obtain an accurate surrogate model.
On the other hand, standard Reduced-Order Models (ROMs) accurately capture
supposedly important physics of the forward model in the reduced subspaces, but
otherwise could be inaccurate elsewhere. In this paper, we propose to enlarge
the validity of ROMs and hence improve the accuracy outside the reduced
subspaces by incorporating a data-driven ML technique. In particular, we focus
on a goal-oriented approach that substantially improves the accuracy of reduced
models by learning the error between the forward model and the ROM outputs.
Once an ML-enhanced ROM is constructed it can accelerate the performance of
solving many-query problems in parametrized forward and inverse problems.
Numerical results for inverse problems governed by elliptic PDEs and
parametrized neutron transport equations will be presented to support our
approach.
