A deep learning approach to solving inverse problems
Author(s)
Jin, Jihui
Advisor(s)
Romberg, Justin K.
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Abstract
The objective of this thesis is to develop machine learning methods to solve inverse problems. Inverse problems arise when there is a signal, image, or volume of interest that can only be measured indirectly. The forward mapping process is typically well understood and can be modeled accurately through analytical methods or simulation. However, inverting the forward model to recover the signal of interest from measurements is typically ill-posed, often requiring extensive computation to recover an acceptable solution.
Modern machine learning methods have achieved tremendous success in recent years by leveraging labeled data (such as signals and their corresponding measurements) to learn arbitrary mappings, typically in a black box manner. Inverse problems offer a rich understanding of the forward mapping process, describing the relationship between the paired data, that can be leveraged for machine learning algorithms. For a class of inverse problems, the forward model is non-linear and expensive to compute, often requiring many iterations to solve a PDE stepping through time and/or space. Computing the gradient (a necessary step for some algorithms) with respect to some input can also be equally as, if not more, expensive to compute. A recurring example throughout this thesis will be ocean acoustic tomography, an application that has many challenging features including a non-linear, expensive, ill-posed forward model as well as limited data.
This research aims to integrate knowledge of the forward model into machine learning solutions to mitigate and address the computational expenses of solving inverse problems. In our first aim, we train a surrogate forward model using supervised learning techniques and integrate it into a classical optimization framework. The surrogate model vastly reduces both the forward and gradient calculation, allowing for cheaper iterates. In the next aim, we improve on this method by incorporating an ensemble of linearizations that approximate the forward model to reduce the black box nature of neural network surrogates. In our third aim, we develop a computationally feasible method to integrate non-linear forward models into "Deep Unrolled" architectures to allow for training end-to-end. We do so by also leveraging an ensemble of linearizations. And finally, in our last aim, we turn our attention towards addressing the high memory costs of training deep unrolled architectures.
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Date
2024-04-27
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Dissertation