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

Accelerating PDE-constrained Inverse Solutions with Deep Learning and Reduced Order Models