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Well-established methods for the solution of stochastic partial differentialequations (SPDEs) typically struggle in problems with high-dimensionalinputs/outputs. Such difficulties are only amplified in large-scaleapplications where even a few tens of full-order model runs are impracticable.While dimensionality reduction can alleviate some of these issues, it is notknown which and how many features of the (high-dimensional) input are actuallypredictive of the (high-dimensional) output. In this paper, we advocate aBayesian formulation that is capable of performing simultaneous dimension andmodel-order reduction. It consists of a component that encodes thehigh-dimensional input into a low-dimensional set of feature functions byemploying sparsity-enforcing priors and a decoding component that makes use ofthe solution of a coarse-grained model in order to reconstruct that of thefull-order model. Both components are represented with latent variables in aprobabilistic graphical model and are simultaneously trained using StochasticVariational Inference methods. The model is capable of quantifying thepredictive uncertainty due to the information loss that unavoidably takes placein any model-order/dimension reduction as well as the uncertainty arising fromfinite-sized training datasets. We demonstrate its capabilities in the contextof random media where fine-scale fluctuations can give rise to random inputswith tens of thousands of variables. With a few tens of full-order modelsimulations, the proposed model is capable of identifying salient physicalfeatures and produce sharp predictions under different boundary conditions ofthe full output which itself consists of thousands of components.

Bayesian model and dimension reduction for uncertainty propagation: applications in random media