Bayesian inference is used extensively to infer and to quantify the uncertainty in a field of interest from a measurement of a related field when the two are linked by a physical model. Despite its many applications, Bayesian inference faces challenges when inferring fields that have discrete representations of large dimension, and or have prior distributions that are difficult to represent mathematically. In this manuscript we consider the use of Generative Adversarial Networks (GANs) in addressing these challenges. A GAN is a type of deep neural network equipped with the ability to learn the distribution implied by multiple samples of a given field. Once trained on these samples, the generator component of a GAN maps the iid components of a low-dimensional latent vector to an approximation of the distribution of the field of interest. In this work we demonstrate how this approximate distribution may be used as a prior in a Bayesian update, and how it addresses the challenges associated with characterizing complex prior distributions and the large dimension of the inferred field. We demonstrate the efficacy of this approach by applying it to the problem of inferring and quantifying uncertainty in the initial temperature field in a heat conduction problem from a noisy measurement of the temperature at later time.
More recently, the approach described in @cite_36 utilizes GANs in a Bayesian setting; however the GAN is trained to approximate the posterior distribution (and not the prior, as in our case), and training is done in a supervised fashion. That is, paired samples of the measurement @math and the corresponding true solution @math are required. In contrast, our approach is unsupervised, where we require only samples of the true solution @math to train the GAN prior.
@cite_36
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Characterizing statistical properties of solutions of inverse problems is essential for decision making. Bayesian inversion offers a tractable framework for this purpose, but current approaches are computationally unfeasible for most realistic imaging applications in the clinic. We introduce two novel deep learning based methods for solving large-scale inverse problems using Bayesian inversion: a sampling based method using a WGAN with a novel mini-discriminator and a direct approach that trains a neural network using a novel loss function. The performance of both methods is demonstrated on image reconstruction in ultra low dose 3D helical CT. We compute the posterior mean and standard deviation of the 3D images followed by a hypothesis test to assess whether a "dark spot" in the liver of a cancer stricken patient is present. Both methods are computationally efficient and our evaluation shows very promising performance that clearly supports the claim that Bayesian inversion is usable for 3D imaging in time critical applications.