Reduced-order model-based variational inference with normalizing flows for Bayesian elliptic inverse problems

Abstract

We propose a reduced-order model-based variational inference method with normalizing flows for Bayesian elliptic inverse problems. The coefficient of the elliptic PDE is represented by a finite number of parameters. We aim to estimate the posterior distributions of these parameters in the framework of variational inference, and the approximation of the posterior distribution is constructed through a normalizing flow. Moreover, as data of inputs and outputs of the forward problem are accumulated during the training of the normalizing flow, we can naturally exploit the low-dimensional intrinsic structure of the forward elliptic PDE using reduced-order models based on these data, in which information of the true posterior is incorporated. We construct a low-dimensional set of data-driven basis functions in the solution space using proper orthogonal decomposition (POD) and train a neural network that maps the parameters to the coefficients of these data-driven basis functions. The surrogate forward map, which is the combination of the reduced-order model and the parameter-to-coefficient neural network, is then applied to the Bayesian elliptic inverse problem, where in each iteration the forward problem is evaluated in the space spanned by the POD basis functions using the surrogate forward map. Thus the inversion will be significantly accelerated as the surrogate forward map is essentially a low-dimensional model. We present numerical examples to show the accuracy and efficiency of the proposed method.