Deep Learning for Forward and Inverse Solutions of Physical Systems

Deep neural networks have recently gained immense popularity, especially for developing surrogate models for applications in the field of science and engineering. These surrogate models are computationally inexpensive and approximate the input-output mapping of the expensive physical simulators. Surrogates can be used efficiently for problems that require expensive, repetitive simulations such as inverse modeling, uncertainty quantification, and propagation or design of experiments tasks. This work focuses on investigating deep learning approaches for the inverse and forward solutions of physical systems. For the first part, a inverse model is developed that maps noisy and gappy observations to a high-dimensional input. This problem is ill-posed since one may not be able to uniquely recover the high-dimensional input given the noisy and sparse output. Traditional inverse modeling techniques require a large number of simulator runs, which is computationally expensive. To address this issue and solve the inverse problem efficiently, we develop an inverse surrogate model that hinges on the concepts of deep generative machine learning and is trained with limited data in a supervised setting. The other part of this work develops an efficient forward surrogate model that is trained with limited data for the forward solution. Here, we develop a data-driven model with a hybrid framework by combining multiscale methods and the deep neural networks for stochastic multiscale PDE systems, in particular, flow in porous media. We then introduce a Bayesian framework to this hybrid model to perform uncertainty quantification and propagation tasks. Overall, deep learning models developed in this work offer a significant acceleration in terms of computational cost and accuracy for both the inverse and forward solutions of physical systems.