A Bayesian Multiscale Deep Learning Framework for Flows in Random Media

Fine-scale simulation of complex systems governed by multiscale partial differential equations (PDEs) is computationally expensive and various multiscale methods have been developed for addressing such problems. In addition, it is challenging to develop accurate surrogate and uncertainty quantification models for high-dimensional problems governed by stochastic multiscale PDEs using limited training data. In this work to address these challenges, we introduce a novel hybrid deep-learning and multiscale approach for stochastic multiscale PDEs with limited training data. For demonstration purposes, we focus on a porous media flow problem. We use an image-to-image supervised deep learning model to learn the mapping between the input permeability field and the multiscale basis functions. We introduce a Bayesian approach to this hybrid framework to allow us to perform uncertainty quantification and propagation tasks. The performance of this hybrid approach is evaluated with varying intrinsic dimensionality of the permeability field. Numerical results indicate that the hybrid network can efficiently predict well for high-dimensional inputs.

Solving inverse problems using conditional invertible neural networks

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 permeability field where the two facies consist of heterogeneous permeability and varying length-scales. For both the two- and three-dimensional surrogate models, the predicted sample realizations of the non-Gaussian permeability field are diverse with the predictive mean being close to the ground truth even when the model is trained with limited data.