Sheroze Sheriffdeen

Sheroze Sheriffdeen, Jean C. Ragusa, Jim E. Morel et al.

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.

Hwan Goh, Sheroze Sheriffdeen, Jonathan Wittmer et al.

In recent years, the field of machine learning has made phenomenal progress in the pursuit of simulating real-world data generation processes. One notable example of such success is the variational autoencoder (VAE). In this work, with a small shift in perspective, we leverage and adapt VAEs for a different purpose: uncertainty quantification in scientific inverse problems. We introduce UQ-VAE: a flexible, adaptive, hybrid data/model-informed framework for training neural networks capable of rapid modelling of the posterior distribution representing the unknown parameter of interest. Specifically, from divergence-based variational inference, our framework is derived such that most of the information usually present in scientific inverse problems is fully utilized in the training procedure. Additionally, this framework includes an adjustable hyperparameter that allows selection of the notion of distance between the posterior model and the target distribution. This introduces more flexibility in controlling how optimization directs the learning of the posterior model. Further, this framework possesses an inherent adaptive optimization property that emerges through the learning of the posterior uncertainty.