Solving inverse problems using conditional invertible neural networks

Highlights: • Developed a surrogate model mapping output information of a physical system to an unknown input distributed parameter. • A generative model based on conditional invertible neural networks (cINN) is developed. • The cINN is trained to serve as an inverse surrogate model of physical systems governed by PDEs. • The inverse surrogate model is used for the solution of inverse problems with unknown spatially-dependent parameters. • The developed method is used to estimate a non-Gaussian permeability field in multiphase flows using limited observations. Inverse modeling for computing a high-dimensional spatially-varying property field from indirect sparse and noisy observations is a challenging problem. This is due to the complex physical system of interest often expressed in the form of multiscale PDEs, the high-dimensionality of the spatial property of interest, and the incomplete and noisy nature of observations. To address these challenges, we develop a model that maps the given observations to the unknown input field in the form of a surrogate model. This inverse surrogate model will then allow us to estimate the unknown input field for any given sparse and noisy output observations. Here, the inverse mapping is limited to a broad prior distribution of the input field with which the surrogate model is trained. In this work, we construct a two- and three-dimensional inverse surrogate models consisting of an invertible and a conditional neural network trained in an end-to-end fashion with limited training data. The invertible network is developed using a flow-based generative model. The developed inverse surrogate model is then applied for an inversion task of a multiphase flow problem where given the pressure and saturation observations the aim is to recover a high-dimensional non-Gaussian log-permeability field where the two facies consist of heterogeneous log-permeability and varying length-scales. For both the two- and three-dimensional surrogate models, the predicted sample realizations of the non-Gaussian log-permeability field are diverse with the predictive mean being close to the ground truth even when the model is trained with limited data.